Theorem List for Metamath Proof Explorer - 44801-44900 *Has distinct variable
group(s)
| Type | Label | Description |
|---|
| Statement |
|---|
| |
| Theorem | uunT1p1 44801 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ ⊤) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT21 44802 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ (𝜑 ∧ 𝜓)) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun121 44803 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun121p1 44804 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun132 44805 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun132p1 44806 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | anabss7p1 44807 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
This would have been named uun221 if the zeroth permutation did not
exist in set.mm as anabss7 673. (Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
| |
| Theorem | un10 44808 |
A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ( ( 𝜑 , ⊤ ) ▶ 𝜓 )
⇒ ⊢ ( 𝜑 ▶ 𝜓 ) |
| |
| Theorem | un01 44809 |
A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ( ( ⊤ , 𝜑 ) ▶ 𝜓 )
⇒ ⊢ ( 𝜑 ▶ 𝜓 ) |
| |
| Theorem | un2122 44810 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun2131 44811 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun2131p1 44812 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜓)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uunTT1 44813 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ ⊤ ∧ 𝜑)
→ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunTT1p1 44814 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ 𝜑 ∧ ⊤)
→ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunTT1p2 44815 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ ⊤ ∧ ⊤)
→ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT11 44816 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ 𝜑 ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT11p1 44817 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ ⊤ ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT11p2 44818 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜑 ∧ ⊤) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | uunT12 44819 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ 𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p1 44820 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((⊤
∧ 𝜓 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p2 44821 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p3 44822 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p4 44823 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓 ∧ ⊤) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uunT12p5 44824 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ 𝜑 ∧ ⊤) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun111 44825 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) |
| |
| Theorem | 3anidm12p1 44826 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
3anidm12 1421 denotes the deduction which would have been
named uun112 if
it did not pre-exist in set.mm. This second permutation's name is based
on this pre-existing name. (Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | 3anidm12p2 44827 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| |
| Theorem | uun123 44828 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun123p1 44829 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun123p2 44830 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜓) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun123p3 44831 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun123p4 44832 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜑) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
| |
| Theorem | uun2221 44833 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 30-Dec-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
| |
| Theorem | uun2221p1 44834 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
| |
| Theorem | uun2221p2 44835 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
(Contributed by Alan Sare, 4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜑 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
| |
| Theorem | 3impdirp1 44836 |
A deduction unionizing a non-unionized collection of virtual hypotheses.
Commuted version of 3impdir 1352. (Contributed by Alan Sare,
4-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜓)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → 𝜃) |
| |
| Theorem | 3impcombi 44837 |
A 1-hypothesis propositional calculus deduction. (Contributed by Alan
Sare, 25-Sep-2017.)
|
| ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃) |
| |
| 21.41.6 Theorems proved using Virtual
Deduction
|
| |
| Theorem | trsspwALT 44838 |
Virtual deduction proof of the left-to-right implication of dftr4 5266. A
transitive class is a subset of its power class. This proof corresponds
to the virtual deduction proof of dftr4 5266 without accumulating results.
(Contributed by Alan Sare, 29-Apr-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
| |
| Theorem | trsspwALT2 44839 |
Virtual deduction proof of trsspwALT 44838. This proof is the same as the
proof of trsspwALT 44838 except each virtual deduction symbol is
replaced by
its non-virtual deduction symbol equivalent. A transitive class is a
subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
| |
| Theorem | trsspwALT3 44840 |
Short predicate calculus proof of the left-to-right implication of
dftr4 5266. A transitive class is a subset of its power
class. This
proof was constructed by applying Metamath's minimize command to the
proof of trsspwALT2 44839, which is the virtual deduction proof trsspwALT 44838
without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
| |
| Theorem | sspwtr 44841 |
Virtual deduction proof of the right-to-left implication of dftr4 5266. A
class which is a subclass of its power class is transitive. This proof
corresponds to the virtual deduction proof of sspwtr 44841 without
accumulating results. (Contributed by Alan Sare, 2-May-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
| |
| Theorem | sspwtrALT 44842 |
Virtual deduction proof of sspwtr 44841. This proof is the same as the
proof of sspwtr 44841 except each virtual deduction symbol is
replaced by
its non-virtual deduction symbol equivalent. A class which is a
subclass of its power class is transitive. (Contributed by Alan Sare,
3-May-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
| |
| Theorem | sspwtrALT2 44843 |
Short predicate calculus proof of the right-to-left implication of
dftr4 5266. A class which is a subclass of its power
class is transitive.
This proof was constructed by applying Metamath's minimize command to
the proof of sspwtrALT 44842, which is the virtual deduction proof sspwtr 44841
without virtual deductions. (Contributed by Alan Sare, 3-May-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
| |
| Theorem | pwtrVD 44844 |
Virtual deduction proof of pwtr 5457; see pwtrrVD 44845 for the converse.
(Contributed by Alan Sare, 25-Aug-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr 𝒫 𝐴) |
| |
| Theorem | pwtrrVD 44845 |
Virtual deduction proof of pwtr 5457; see pwtrVD 44844 for the converse.
(Contributed by Alan Sare, 25-Aug-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ (Tr 𝒫 𝐴 → Tr 𝐴) |
| |
| Theorem | suctrALT 44846 |
The successor of a transitive class is transitive. The proof of
https://us.metamath.org/other/completeusersproof/suctrvd.html
is a
Virtual Deduction proof verified by automatically transforming it into
the Metamath proof of suctrALT 44846 using completeusersproof, which is
verified by the Metamath program. The proof of
https://us.metamath.org/other/completeusersproof/suctrro.html 44846 is a
form of the completed proof which preserves the Virtual Deduction
proof's step numbers and their ordering. See suctr 6470 for the original
proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare,
12-Jun-2018.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | snssiALTVD 44847 |
Virtual deduction proof of snssiALT 44848. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
| |
| Theorem | snssiALT 44848 |
If a class is an element of another class, then its singleton is a
subclass of that other class. Alternate proof of snssi 4808. This
theorem was automatically generated from snssiALTVD 44847 using a
translation program. (Contributed by Alan Sare, 11-Sep-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
| |
| Theorem | snsslVD 44849 |
Virtual deduction proof of snssl 44850. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
| |
| Theorem | snssl 44850 |
If a singleton is a subclass of another class, then the singleton's
element is an element of that other class. This theorem is the
right-to-left implication of the biconditional snss 4785.
The proof of
this theorem was automatically generated from snsslVD 44849 using a tools
command file, translateMWO.cmd, by translating the proof into its
non-virtual deduction form and minimizing it. (Contributed by Alan
Sare, 25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵) |
| |
| Theorem | snelpwrVD 44851 |
Virtual deduction proof of snelpwi 5448. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
| |
| Theorem | unipwrVD 44852 |
Virtual deduction proof of unipwr 44853. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
| |
| Theorem | unipwr 44853 |
A class is a subclass of the union of its power class. This theorem is
the right-to-left subclass lemma of unipw 5455. The proof of this theorem
was automatically generated from unipwrVD 44852 using a tools command file ,
translateMWO.cmd , by translating the proof into its non-virtual
deduction form and minimizing it. (Contributed by Alan Sare,
25-Aug-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
| |
| Theorem | sstrALT2VD 44854 |
Virtual deduction proof of sstrALT2 44855. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
| |
| Theorem | sstrALT2 44855 |
Virtual deduction proof of sstr 3992, transitivity of subclasses, Theorem
6 of [Suppes] p. 23. This theorem was
automatically generated from
sstrALT2VD 44854 using the command file
translate_without_overwriting.cmd . It was not minimized because the
automated minimization excluding duplicates generates a minimized proof
which, although not directly containing any duplicates, indirectly
contains a duplicate. That is, the trace back of the minimized proof
contains a duplicate. This is undesirable because some step(s) of the
minimized proof use the proven theorem. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
| |
| Theorem | suctrALT2VD 44856 |
Virtual deduction proof of suctrALT2 44857. (Contributed by Alan Sare,
11-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | suctrALT2 44857 |
Virtual deduction proof of suctr 6470. The successor of a transitive
class is transitive. This proof was generated automatically from the
virtual deduction proof suctrALT2VD 44856 using the tools command file
translate_without_overwriting_minimize_excluding_duplicates.cmd .
(Contributed by Alan Sare, 11-Sep-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | elex2VD 44858* |
Virtual deduction proof of elex2 2818. (Contributed by Alan Sare,
25-Sep-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
| |
| Theorem | elex22VD 44859* |
Virtual deduction proof of elex22 3506. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) |
| |
| Theorem | eqsbc2VD 44860* |
Virtual deduction proof of eqsbc2 3854. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴)) |
| |
| Theorem | zfregs2VD 44861* |
Virtual deduction proof of zfregs2 9773. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ≠ ∅ → ¬
∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) |
| |
| Theorem | tpid3gVD 44862 |
Virtual deduction proof of tpid3g 4772. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
| |
| Theorem | en3lplem1VD 44863* |
Virtual deduction proof of en3lplem1 9652. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))) |
| |
| Theorem | en3lplem2VD 44864* |
Virtual deduction proof of en3lplem2 9653. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))) |
| |
| Theorem | en3lpVD 44865 |
Virtual deduction proof of en3lp 9654. (Contributed by Alan Sare,
24-Oct-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) |
| |
| 21.41.7 Theorems proved using Virtual Deduction
with mmj2 assistance
|
| |
| Theorem | simplbi2VD 44866 |
Virtual deduction proof of simplbi2 500. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
| | 3:1,?: e0a 44792 | ⊢ ((𝜓 ∧ 𝜒) → 𝜑)
| | qed:3,?: e0a 44792 | ⊢ (𝜓 → (𝜒 → 𝜑))
|
The proof of simplbi2 500 was automatically derived from it.
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 → (𝜒 → 𝜑)) |
| |
| Theorem | 3ornot23VD 44867 |
Virtual deduction proof of 3ornot23 44529. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ (¬ 𝜑
∧ ¬ 𝜓) )
| | 2:: | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ (𝜒 ∨ 𝜑 ∨ 𝜓) )
| | 3:1,?: e1a 44647 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜑 )
| | 4:1,?: e1a 44647 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ 𝜓 )
| | 5:3,4,?: e11 44708 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ¬ (𝜑
∨ 𝜓) )
| | 6:2,?: e2 44651 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ (𝜒 ∨ (𝜑 ∨ 𝜓)) )
| | 7:5,6,?: e12 44744 | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) , (𝜒 ∨ 𝜑
∨ 𝜓) ▶ 𝜒 )
| | 8:7: | ⊢ ( (¬ 𝜑 ∧ ¬ 𝜓) ▶ ((𝜒
∨ 𝜑 ∨ 𝜓) → 𝜒) )
| | qed:8: | ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒
∨ 𝜑 ∨ 𝜓) → 𝜒))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒)) |
| |
| Theorem | orbi1rVD 44868 |
Virtual deduction proof of orbi1r 44530. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| | 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜒 ∨ 𝜑) )
| | 3:2,?: e2 44651 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜑 ∨ 𝜒) )
| | 4:1,3,?: e12 44744 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜓 ∨ 𝜒) )
| | 5:4,?: e2 44651 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜑)
▶ (𝜒 ∨ 𝜓) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜑)
→ (𝜒 ∨ 𝜓)) )
| | 7:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜓) )
| | 8:7,?: e2 44651 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜓 ∨ 𝜒) )
| | 9:1,8,?: e12 44744 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜑 ∨ 𝜒) )
| | 10:9,?: e2 44651 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ∨ 𝜓)
▶ (𝜒 ∨ 𝜑) )
| | 11:10: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ∨ 𝜓)
→ (𝜒 ∨ 𝜑)) )
| | 12:6,11,?: e11 44708 | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒
∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) )
| | qed:12: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑)
↔ (𝜒 ∨ 𝜓)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))) |
| |
| Theorem | bitr3VD 44869 |
Virtual deduction proof of bitr3 352. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑
↔ 𝜓) )
| | 2:1,?: e1a 44647 | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜓
↔ 𝜑) )
| | 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜑 ↔ 𝜒) )
| | 4:3,?: e2 44651 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜒 ↔ 𝜑) )
| | 5:2,4,?: e12 44744 | ⊢ ( (𝜑 ↔ 𝜓) , (𝜑 ↔ 𝜒)
▶ (𝜓 ↔ 𝜒) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜑
↔ 𝜒) → (𝜓 ↔ 𝜒)) )
| | qed:6: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒)
→ (𝜓 ↔ 𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒))) |
| |
| Theorem | 3orbi123VD 44870 |
Virtual deduction proof of 3orbi123 44531. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧
(𝜏 ↔ 𝜂)) )
| | 2:1,?: e1a 44647 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜑 ↔ 𝜓) )
| | 3:1,?: e1a 44647 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜒 ↔ 𝜃) )
| | 4:1,?: e1a 44647 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (𝜏 ↔ 𝜂) )
| | 5:2,3,?: e11 44708 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) )
| | 6:5,4,?: e11 44708 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 7:?: | ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ (𝜑
∨ 𝜒 ∨ 𝜏))
| | 8:6,7,?: e10 44714 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃)
∨ 𝜂)) )
| | 9:?: | ⊢ (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔
(𝜓 ∨ 𝜃 ∨ 𝜂))
| | 10:8,9,?: e10 44714 | ⊢ ( ((𝜑 ↔ 𝜓) ∧ (𝜒
↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) ▶ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨
𝜃 ∨ 𝜂)) )
| | qed:10: | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)
∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃
∨ 𝜂)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))) |
| |
| Theorem | sbc3orgVD 44871 |
Virtual deduction proof of the analogue of sbcor 3839 with three disjuncts.
The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:1,?: e1a 44647 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝜑
∨ 𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 3:: | ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑
∨ 𝜓 ∨ 𝜒))
| | 32:3: | ⊢ ∀𝑥(((𝜑 ∨ 𝜓) ∨ 𝜒)
↔ (𝜑 ∨ 𝜓 ∨ 𝜒))
| | 33:1,32,?: e10 44714 | ⊢ ( 𝐴 ∈ 𝐵 ▶ [𝐴 / 𝑥](((𝜑
∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 4:1,33,?: e11 44708 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝜑
∨ 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒)) )
| | 5:2,4,?: e11 44708 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒)) )
| | 6:1,?: e1a 44647 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) )
| | 7:6,?: e1a 44647 | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥](𝜑
∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 8:5,7,?: e11 44708 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)
∨ [𝐴 / 𝑥]𝜒)) )
| | 9:?: | ⊢ ((([𝐴 / 𝑥]𝜑
∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑
∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))
| | 10:8,9,?: e10 44714 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓
∨ [𝐴 / 𝑥]𝜒)) )
| | qed:10: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑
∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓
∨ [𝐴 / 𝑥]𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))) |
| |
| Theorem | 19.21a3con13vVD 44872* |
Virtual deduction proof of alrim3con13v 44553. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 → ∀𝑥𝜑)
▶ (𝜑 → ∀𝑥𝜑) )
| | 2:: | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓 ∧ 𝜑
∧ 𝜒) ▶ (𝜓 ∧ 𝜑 ∧ 𝜒) )
| | 3:2,?: e2 44651 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜓 )
| | 4:2,?: e2 44651 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜑 )
| | 5:2,?: e2 44651 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ 𝜒 )
| | 6:1,4,?: e12 44744 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜑 )
| | 7:3,?: e2 44651 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜓 )
| | 8:5,?: e2 44651 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥𝜒 )
| | 9:7,6,8,?: e222 44656 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒) )
| | 10:9,?: e2 44651 | ⊢ ( (𝜑 → ∀𝑥𝜑) , (𝜓
∧ 𝜑 ∧ 𝜒) ▶ ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒) )
| | 11:10:in2 | ⊢ ( (𝜑 → ∀𝑥𝜑) ▶ ((𝜓
∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)) )
| | qed:11:in1 | ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓
∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒))) |
| |
| Theorem | exbirVD 44873 |
Virtual deduction proof of exbir 44499. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
▶ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) )
| | 2:: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ,
(𝜑 ∧ 𝜓) ▶ (𝜑 ∧ 𝜓) )
| | 3:: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ,
(𝜑 ∧ 𝜓), 𝜃 ▶ 𝜃 )
| | 5:1,2,?: e12 44744 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓) ▶ (𝜒 ↔ 𝜃) )
| | 6:3,5,?: e32 44778 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓), 𝜃 ▶ 𝜒 )
| | 7:6: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)), (𝜑 ∧ 𝜓) ▶ (𝜃 → 𝜒) )
| | 8:7: | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
▶ ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒)) )
| | 9:8,?: e1a 44647 | ⊢ ( ((𝜑 ∧ 𝜓) → (𝜒
↔ 𝜃)) ▶ (𝜑 → (𝜓 → (𝜃 → 𝜒))) )
| | qed:9: | ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
→ (𝜑 → (𝜓 → (𝜃 → 𝜒))))
|
(Contributed by Alan Sare, 13-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) |
| |
| Theorem | exbiriVD 44874 |
Virtual deduction proof of exbiri 811. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
| | 2:: | ⊢ ( 𝜑 ▶ 𝜑 )
| | 3:: | ⊢ ( 𝜑 , 𝜓 ▶ 𝜓 )
| | 4:: | ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜃 )
| | 5:2,1,?: e10 44714 | ⊢ ( 𝜑 ▶ (𝜓 → (𝜒 ↔ 𝜃)) )
| | 6:3,5,?: e21 44750 | ⊢ ( 𝜑 , 𝜓 ▶ (𝜒 ↔ 𝜃) )
| | 7:4,6,?: e32 44778 | ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜒 )
| | 8:7: | ⊢ ( 𝜑 , 𝜓 ▶ (𝜃 → 𝜒) )
| | 9:8: | ⊢ ( 𝜑 ▶ (𝜓 → (𝜃 → 𝜒)) )
| | qed:9: | ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) |
| |
| Theorem | rspsbc2VD 44875* |
Virtual deduction proof of rspsbc2 44554. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:: | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ 𝐶 ∈ 𝐷 )
| | 3:: | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 )
| | 4:1,3,?: e13 44768 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑 )
| | 5:1,4,?: e13 44768 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑 )
| | 6:2,5,?: e23 44775 | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑 )
| | 7:6: | ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ (∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑) )
| | 8:7: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝐶 ∈ 𝐷
→ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) )
| | qed:8: | ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷
→ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))) |
| |
| Theorem | 3impexpVD 44876 |
Virtual deduction proof of 3impexp 1359. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) )
| | 2:: | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒)
↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
| | 3:1,2,?: e10 44714 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 4:3,?: e1a 44647 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 5:4,?: e1a 44647 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ▶ (𝜑 → (𝜓 → (𝜒 → 𝜃))) )
| | 6:5: | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
→ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
| | 7:: | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ (𝜑 → (𝜓 → (𝜒 → 𝜃))) )
| | 8:7,?: e1a 44647 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) )
| | 9:8,?: e1a 44647 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) )
| | 10:2,9,?: e01 44711 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ 𝜃))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) )
| | 11:10: | ⊢ ((𝜑 → (𝜓 → (𝜒
→ 𝜃))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
| | qed:6,11,?: e00 44788 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒)
→ 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃)))) |
| |
| Theorem | 3impexpbicomVD 44877 |
Virtual deduction proof of 3impexpbicom 44500. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) )
| | 2:: | ⊢ ((𝜃 ↔ 𝜏) ↔ (𝜏
↔ 𝜃))
| | 3:1,2,?: e10 44714 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃)) )
| | 4:3,?: e1a 44647 | ⊢ ( ((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ▶ (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))) )
| | 5:4: | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) → (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))))
| | 6:: | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))) )
| | 7:6,?: e1a 44647 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏
↔ 𝜃)) )
| | 8:7,2,?: e10 44714 | ⊢ ( (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) ▶ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃
↔ 𝜏)) )
| | 9:8: | ⊢ ((𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃)))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃
↔ 𝜏)))
| | qed:5,9,?: e00 44788 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒)
→ (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏
↔ 𝜃)))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))) |
| |
| Theorem | 3impexpbicomiVD 44878 |
Virtual deduction proof of 3impexpbicomi 44501. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃
↔ 𝜏))
| | qed:1,?: e0a 44792 | ⊢ (𝜑 → (𝜓 → (𝜒
→ (𝜏 ↔ 𝜃))))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))) |
| |
| Theorem | sbcoreleleqVD 44879* |
Virtual deduction proof of sbcoreleleq 44555. The following user's proof
is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:1,?: e1a 44647 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 ∈
𝑦 ↔ 𝑥 ∈ 𝐴) )
| | 3:1,?: e1a 44647 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑦 ∈
𝑥 ↔ 𝐴 ∈ 𝑥) )
| | 4:1,?: e1a 44647 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦]𝑥 =
𝑦 ↔ 𝑥 = 𝐴) )
| | 5:2,3,4,?: e111 44694 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ((𝑥 ∈ 𝐴
∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥
∨ [𝐴 / 𝑦]𝑥 = 𝑦)) )
| | 6:1,?: e1a 44647 | ⊢ ( 𝐴 ∈ 𝐵
▶ ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥
∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥 ∨ [𝐴 / 𝑦]𝑥 = 𝑦)) )
| | 7:5,6: e11 44708 | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑦](𝑥
∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)) )
| | qed:7: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴)))
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴))) |
| |
| Theorem | hbra2VD 44880* |
Virtual deduction proof of nfra2 3376. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ (∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑 →
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 2:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 3:1,2,?: e00 44788 | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 →
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 4:2: | ⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | 5:4,?: e0a 44792 | ⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔
∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
| | qed:3,5,?: e00 44788 | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 →
∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑)
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) |
| |
| Theorem | tratrbVD 44881* |
Virtual deduction proof of tratrb 44556. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)
∧ 𝐵 ∈ 𝐴) )
| | 2:1,?: e1a 44647 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| | 3:1,?: e1a 44647 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 4:1,?: e1a 44647 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ∈ 𝐴 )
| | 5:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) )
| | 6:5,?: e2 44651 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝑦 )
| | 7:5,?: e2 44651 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑦 ∈ 𝐵 )
| | 8:2,7,4,?: e121 44676 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑦 ∈ 𝐴 )
| | 9:2,6,8,?: e122 44673 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝐴 )
| | 10:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥 ▶ 𝐵 ∈ 𝑥 )
| | 11:6,7,10,?: e223 44655 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥 ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥) )
| | 12:11: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝐵 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥)) )
| | 13:: | ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵
∧ 𝐵 ∈ 𝑥)
| | 14:12,13,?: e20 44747 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝐵 ∈ 𝑥 )
| | 15:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ 𝑥 = 𝐵 )
| | 16:7,15,?: e23 44775 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ 𝑦 ∈ 𝑥 )
| | 17:6,16,?: e23 44775 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) )
| | 18:17: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 = 𝐵 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)) )
| | 19:: | ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)
| | 20:18,19,?: e20 44747 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝑥 = 𝐵 )
| | 21:3,?: e1a 44647 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑦 ∈ 𝐴
∀𝑥 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 22:21,9,4,?: e121 44676 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦) )
| | 23:22,?: e2 44651 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ [𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 24:4,23,?: e12 44744 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵) )
| | 25:14,20,24,?: e222 44656 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝐵 )
| | 26:25: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ((𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| | 27:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨
𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 28:27,?: e0a 44792 | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
→ ∀𝑦(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
| | 29:28,26: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| | 30:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑥∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 31:30,?: e0a 44792 | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑥(Tr 𝐴
∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
| | 32:31,29: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥
∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| | 33:32,?: e1a 44647 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐵 )
| | qed:33: | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵) |
| |
| Theorem | al2imVD 44882 |
Virtual deduction proof of al2im 1814. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ ∀𝑥(𝜑 → (𝜓 → 𝜒)) )
| | 2:1,?: e1a 44647 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)) )
| | 3:: | ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓
→ ∀𝑥𝜒))
| | 4:2,3,?: e10 44714 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) )
| | qed:4: | ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒))
→ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))) |
| |
| Theorem | syl5impVD 44883 |
Virtual deduction proof of syl5imp 44532. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
| 1:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜑
→ (𝜓 → 𝜒)) )
| | 2:1,?: e1a 44647 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜓
→ (𝜑 → 𝜒)) )
| | 3:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → 𝜓) )
| | 4:3,2,?: e21 44750 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → (𝜑 → 𝜒)) )
| | 5:4,?: e2 44651 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜑 → (𝜃 → 𝜒)) )
| | 6:5: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ ((𝜃
→ 𝜓) → (𝜑 → (𝜃 → 𝜒))) )
| | qed:6: | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃
→ 𝜓) → (𝜑 → (𝜃 → 𝜒))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒)))) |
| |
| Theorem | idiVD 44884 |
Virtual deduction proof of idiALT 44498. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
| h1:: | ⊢ 𝜑
| | qed:1,?: e0a 44792 | ⊢ 𝜑
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝜑 ⇒ ⊢ 𝜑 |
| |
| Theorem | ancomstVD 44885 |
Closed form of ancoms 458. The following user's proof is completed by
invoking mmj2's unify command and using mmj2's StepSelector to pick all
remaining steps of the Metamath proof.
| 1:: | ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
| | qed:1,?: e0a 44792 | ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓
∧ 𝜑) → 𝜒))
|
The proof of ancomst 464 is derived automatically from it.
(Contributed by
Alan Sare, 25-Dec-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒)) |
| |
| Theorem | ssralv2VD 44886* |
Quantification restricted to a subclass for two quantifiers. ssralv 4052
for two quantifiers. The following User's Proof is a Virtual Deduction
proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. ssralv2 44551 is ssralv2VD 44886 without
virtual deductions and was automatically derived from ssralv2VD 44886.
| 1:: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ▶ (𝐴 ⊆ 𝐵
∧ 𝐶 ⊆ 𝐷) )
| | 2:: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 )
| | 3:1: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ▶ 𝐴 ⊆ 𝐵 )
| | 4:3,2: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐷𝜑 )
| | 5:4: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑) )
| | 6:5: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑) )
| | 7:: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 )
| | 8:7,6: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴 ▶ ∀𝑦 ∈ 𝐷𝜑 )
| | 9:1: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ▶ 𝐶 ⊆ 𝐷 )
| | 10:9,8: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴 ▶ ∀𝑦 ∈ 𝐶𝜑 )
| | 11:10: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑) )
| | 12:: | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
→ ∀𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷))
| | 13:: | ⊢ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑
→ ∀𝑥∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑)
| | 14:12,13,11: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑) )
| | 15:14: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑 )
| | 16:15: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
▶ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑) )
| | qed:16: | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
→ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑))
|
(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)) |
| |
| Theorem | ordelordALTVD 44887 |
An element of an ordinal class is ordinal. Proposition 7.6 of
[TakeutiZaring] p. 36. This is an alternate proof of ordelord 6406 using
the Axiom of Regularity indirectly through dford2 9660. dford2 is a
weaker definition of ordinal number. Given the Axiom of Regularity, it
need not be assumed that E Fr 𝐴 because this is inferred by the
Axiom of Regularity. The following User's Proof is a Virtual Deduction
proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. ordelordALT 44557 is ordelordALTVD 44887
without virtual deductions and was automatically derived from
ordelordALTVD 44887 using the tools program
translate..without..overwriting.cmd and the Metamath program "MM-PA>
MINIMIZE_WITH *" command.
| 1:: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ (Ord 𝐴
∧ 𝐵 ∈ 𝐴) )
| | 2:1: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Ord 𝐴 )
| | 3:1: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ∈ 𝐴 )
| | 4:2: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| | 5:2: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) )
| | 6:4,3: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ⊆ 𝐴 )
| | 7:6,6,5: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐵(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) )
| | 8:: | ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)
↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 9:8: | ⊢ ∀𝑦((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)
↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 10:9: | ⊢ ∀𝑦 ∈ 𝐴((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦
∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 11:10: | ⊢ (∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦
∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 12:11: | ⊢ ∀𝑥(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦
∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 13:12: | ⊢ ∀𝑥 ∈ 𝐴(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| | 14:13: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦))
| | 15:14,5: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| | 16:4,15,3: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐵 )
| | 17:16,7: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Ord 𝐵 )
| | qed:17: | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
|
(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) |
| |
| Theorem | equncomVD 44888 |
If a class equals the union of two other classes, then it equals the union
of those two classes commuted. The following User's Proof is a Virtual
Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. equncom 4159 is equncomVD 44888 without
virtual deductions and was automatically derived from equncomVD 44888.
| 1:: | ⊢ ( 𝐴 = (𝐵 ∪ 𝐶) ▶ 𝐴 = (𝐵 ∪ 𝐶) )
| | 2:: | ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
| | 3:1,2: | ⊢ ( 𝐴 = (𝐵 ∪ 𝐶) ▶ 𝐴 = (𝐶 ∪ 𝐵) )
| | 4:3: | ⊢ (𝐴 = (𝐵 ∪ 𝐶) → 𝐴 = (𝐶 ∪ 𝐵))
| | 5:: | ⊢ ( 𝐴 = (𝐶 ∪ 𝐵) ▶ 𝐴 = (𝐶 ∪ 𝐵) )
| | 6:5,2: | ⊢ ( 𝐴 = (𝐶 ∪ 𝐵) ▶ 𝐴 = (𝐵 ∪ 𝐶) )
| | 7:6: | ⊢ (𝐴 = (𝐶 ∪ 𝐵) → 𝐴 = (𝐵 ∪ 𝐶))
| | 8:4,7: | ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))
|
(Contributed by Alan Sare, 17-Feb-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) |
| |
| Theorem | equncomiVD 44889 |
Inference form of equncom 4159. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. equncomi 4160 is equncomiVD 44889 without
virtual deductions and was automatically derived from equncomiVD 44889.
| h1:: | ⊢ 𝐴 = (𝐵 ∪ 𝐶)
| | qed:1: | ⊢ 𝐴 = (𝐶 ∪ 𝐵)
|
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐴 = (𝐶 ∪ 𝐵) |
| |
| Theorem | sucidALTVD 44890 |
A set belongs to its successor. Alternate proof of sucid 6466.
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. sucidALT 44891 is sucidALTVD 44890
without virtual deductions and was automatically derived from
sucidALTVD 44890. This proof illustrates that
completeusersproof.cmd will generate a Metamath proof from any
User's Proof which is "conventional" in the sense that no step
is a virtual deduction, provided that all necessary unification
theorems and transformation deductions are in set.mm.
completeusersproof.cmd automatically converts such a
conventional proof into a Virtual Deduction proof for which each
step happens to be a 0-virtual hypothesis virtual deduction.
The user does not need to search for reference theorem labels or
deduction labels nor does he(she) need to use theorems and
deductions which unify with reference theorems and deductions in
set.mm. All that is necessary is that each theorem or deduction
of the User's Proof unifies with some reference theorem or
deduction in set.mm or is a semantic variation of some theorem
or deduction which unifies with some reference theorem or
deduction in set.mm. The definition of "semantic variation" has
not been precisely defined. If it is obvious that a theorem or
deduction has the same meaning as another theorem or deduction,
then it is a semantic variation of the latter theorem or
deduction. For example, step 4 of the User's Proof is a
semantic variation of the definition (axiom)
suc 𝐴 = (𝐴 ∪ {𝐴}), which unifies with df-suc 6390, a
reference definition (axiom) in set.mm. Also, a theorem or
deduction is said to be a semantic variation of another
theorem or deduction if it is obvious upon cursory inspection
that it has the same meaning as a weaker form of the latter
theorem or deduction. For example, the deduction Ord 𝐴
infers ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) is a
semantic variation of the theorem (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))), which unifies with
the set.mm reference definition (axiom) dford2 9660.
| h1:: | ⊢ 𝐴 ∈ V
| | 2:1: | ⊢ 𝐴 ∈ {𝐴}
| | 3:2: | ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴)
| | 4:: | ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴)
| | qed:3,4: | ⊢ 𝐴 ∈ suc 𝐴
|
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ 𝐴 ∈ suc 𝐴 |
| |
| Theorem | sucidALT 44891 |
A set belongs to its successor. This proof was automatically derived
from sucidALTVD 44890 using translate_without_overwriting.cmd and
minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ 𝐴 ∈ suc 𝐴 |
| |
| Theorem | sucidVD 44892 |
A set belongs to its successor. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools
program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant.
sucid 6466 is sucidVD 44892 without virtual deductions and was automatically
derived from sucidVD 44892.
| h1:: | ⊢ 𝐴 ∈ V
| | 2:1: | ⊢ 𝐴 ∈ {𝐴}
| | 3:2: | ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴})
| | 4:: | ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
| | qed:3,4: | ⊢ 𝐴 ∈ suc 𝐴
|
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ 𝐴 ∈ suc 𝐴 |
| |
| Theorem | imbi12VD 44893 |
Implication form of imbi12i 350. The following User's Proof is a Virtual
Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. imbi12 346 is imbi12VD 44893 without virtual
deductions and was automatically derived from imbi12VD 44893.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| | 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ (𝜒 ↔ 𝜃) )
| | 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜑
→ 𝜒) ▶ (𝜑 → 𝜒) )
| | 4:1,3: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜑
→ 𝜒) ▶ (𝜓 → 𝜒) )
| | 5:2,4: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜑
→ 𝜒) ▶ (𝜓 → 𝜃) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜑 → 𝜒) → (𝜓 → 𝜃)) )
| | 7:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜓
→ 𝜃) ▶ (𝜓 → 𝜃) )
| | 8:1,7: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜓
→ 𝜃) ▶ (𝜑 → 𝜃) )
| | 9:2,8: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜓
→ 𝜃) ▶ (𝜑 → 𝜒) )
| | 10:9: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜓 → 𝜃) → (𝜑 → 𝜒)) )
| | 11:6,10: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)) )
| | 12:11: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ↔ 𝜃)
→ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) )
| | qed:12: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃)
→ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))
|
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))) |
| |
| Theorem | imbi13VD 44894 |
Join three logical equivalences to form equivalence of implications. The
following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi13 44540
is imbi13VD 44894 without virtual deductions and was automatically derived
from imbi13VD 44894.
| 1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| | 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ (𝜒 ↔ 𝜃) )
| | 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ (𝜏 ↔ 𝜂) )
| | 4:2,3: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂)) )
| | 5:1,4: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))) )
| | 6:5: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂)))) )
| | 7:6: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ↔ 𝜃)
→ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂))))) )
| | qed:7: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃)
→ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂))))))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))) |
| |
| Theorem | sbcim2gVD 44895 |
Distribution of class substitution over a left-nested implication.
Similar to sbcimg 3837.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
sbcim2g 44558 is sbcim2gVD 44895 without virtual deductions and was automatically
derived from sbcim2gVD 44895.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:: | ⊢ ( 𝐴 ∈ 𝐵 , [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ▶ [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) )
| | 3:1,2: | ⊢ ( 𝐴 ∈ 𝐵 , [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ▶ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) )
| | 4:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜓 → 𝜒)
↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) )
| | 5:3,4: | ⊢ ( 𝐴 ∈ 𝐵 , [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ▶ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒)) )
| | 6:5: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒))) )
| | 7:: | ⊢ ( 𝐴 ∈ 𝐵 , ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) ▶ ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) )
| | 8:4,7: | ⊢ ( 𝐴 ∈ 𝐵 , ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) ▶ ([𝐴 / 𝑥]𝜑
→ [𝐴 / 𝑥](𝜓 → 𝜒)) )
| | 9:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))) )
| | 10:8,9: | ⊢ ( 𝐴 ∈ 𝐵 , ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) ▶ [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) )
| | 11:10: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒))) )
| | 12:6,11: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
→ (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒))) )
| | qed:12: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒))))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) |
| |
| Theorem | sbcbiVD 44896 |
Implication form of sbcbii 3846.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
sbcbi 44559 is sbcbiVD 44896 without virtual deductions and was automatically
derived from sbcbiVD 44896.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:: | ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓)
▶ ∀𝑥(𝜑 ↔ 𝜓) )
| | 3:1,2: | ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓)
▶ [𝐴 / 𝑥](𝜑 ↔ 𝜓) )
| | 4:1,3: | ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓)
▶ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) )
| | 5:4: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑥(𝜑 ↔ 𝜓)
→ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) )
| | qed:5: | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓)
→ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
| |
| Theorem | trsbcVD 44897* |
Formula-building inference rule for class substitution, substituting a
class variable for the setvar variable of the transitivity predicate.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
trsbc 44560 is trsbcVD 44897 without virtual deductions and was automatically
derived from trsbcVD 44897.
| 1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| | 2:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑧 ∈ 𝑦
↔ 𝑧 ∈ 𝑦) )
| | 3:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝑥
↔ 𝑦 ∈ 𝐴) )
| | 4:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑧 ∈ 𝑥
↔ 𝑧 ∈ 𝐴) )
| | 5:1,2,3,4: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝑧 ∈ 𝑦
→ ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))) )
| | 6:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 →
([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥))) )
| | 7:5,6: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴
→ 𝑧 ∈ 𝐴))) )
| | 8:: | ⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴
→ 𝑧 ∈ 𝐴)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
| | 9:7,8: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 10:: | ⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥
→ 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
| | 11:10: | ⊢ ∀𝑥((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥
→ 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
| | 12:1,11: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)
→ 𝑧 ∈ 𝑥)) )
| | 13:9,12: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 14:13: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑦([𝐴 / 𝑥]((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 15:14: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑦[𝐴 / 𝑥]((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 16:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) )
| | 17:15,16: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 18:17: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑧([𝐴 / 𝑥]∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| | 19:18: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑧[𝐴 / 𝑥]∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) )
| | 20:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑧∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) )
| | 21:19,20: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑧∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) )
| | 22:: | ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
| | 23:21,22: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑧∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴) )
| | 24:: | ⊢ (Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦
∈ 𝑥) → 𝑧 ∈ 𝑥))
| | 25:24: | ⊢ ∀𝑥(Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
| | 26:1,25: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]Tr 𝑥
↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) )
| | 27:23,26: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]Tr 𝑥
↔ Tr 𝐴) )
| | qed:27: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥
↔ Tr 𝐴))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴)) |
| |
| Theorem | truniALTVD 44898* |
The union of a class of transitive sets is transitive.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
truniALT 44561 is truniALTVD 44898 without virtual deductions and was
automatically derived from truniALTVD 44898.
| 1:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ∀𝑥 ∈ 𝐴
Tr 𝑥 )
| | 2:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) )
| | 3:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ 𝑧 ∈ 𝑦 )
| | 4:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ 𝑦 ∈ ∪ 𝐴 )
| | 5:4: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) )
| | 6:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) )
| | 7:6: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑦 ∈ 𝑞 )
| | 8:6: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑞 ∈ 𝐴 )
| | 9:1,8: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ [𝑞 / 𝑥]Tr 𝑥 )
| | 10:8,9: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ Tr 𝑞 )
| | 11:3,7,10: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑧 ∈ 𝑞 )
| | 12:11,8: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑧 ∈ ∪ 𝐴 )
| | 13:12: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 14:13: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 15:14: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 16:5,15: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ 𝑧 ∈ ∪ 𝐴 )
| | 17:16: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ((𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 18:17: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥
▶ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| | 19:18: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ Tr ∪ 𝐴 )
| | qed:19: | ⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∪ 𝐴)
|
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪
𝐴) |
| |
| Theorem | ee33VD 44899 |
Non-virtual deduction form of e33 44754.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
ee33 44541 is ee33VD 44899 without virtual deductions and was automatically
derived from ee33VD 44899.
| h1:: | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
| | h2:: | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
| | h3:: | ⊢ (𝜃 → (𝜏 → 𝜂))
| | 4:1,3: | ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
| | 5:4: | ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
| | 6:2,5: | ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓
→ (𝜒 → 𝜂))))))
| | 7:6: | ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒
→ 𝜂)))))
| | 8:7: | ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
| | qed:8: | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
|
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) |
| |
| Theorem | trintALTVD 44900* |
The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 44901.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
trintALT 44901 is trintALTVD 44900 without virtual deductions and was
automatically derived from trintALTVD 44900.
| 1:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ∀𝑥 ∈ 𝐴Tr 𝑥 )
| | 2:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) )
| | 3:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ 𝑧 ∈ 𝑦 )
| | 4:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ 𝑦 ∈ ∩ 𝐴 )
| | 5:4: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ ∀𝑞 ∈ 𝐴𝑦 ∈ 𝑞 )
| | 6:5: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞) )
| | 7:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ 𝑞 ∈ 𝐴 )
| | 8:7,6: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ 𝑦 ∈ 𝑞 )
| | 9:7,1: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ [𝑞 / 𝑥]Tr 𝑥 )
| | 10:7,9: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ Tr 𝑞 )
| | 11:10,3,8: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ 𝑧 ∈ 𝑞 )
| | 12:11: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞) )
| | 13:12: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞) )
| | 14:13: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ ∀𝑞 ∈ 𝐴𝑧 ∈ 𝑞 )
| | 15:3,14: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ 𝑧 ∈ ∩ 𝐴 )
| | 16:15: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ((𝑧 ∈ 𝑦 ∧ 𝑦
∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴) )
| | 17:16: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ∀𝑧∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴) )
| | 18:17: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ Tr ∩ 𝐴 )
| | qed:18: | ⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∩ 𝐴)
|
(Contributed by Alan Sare, 17-Apr-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩
𝐴) |
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