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Theorem List for Metamath Proof Explorer - 38801-38900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempetinidres2 38801 Class 𝐴 is a partition by an intersection with the identity class restricted to it if and only if the cosets by the intersection are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
(( Disj (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( I ↾ 𝐴)) / (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( I ↾ 𝐴)) / ≀ (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴))
 
Theorempetinidres 38802 A class is a partition by an intersection with the identity class restricted to it if and only if the cosets by the intersection are in equivalence relation on it. Cf. br1cossinidres 38430, disjALTVinidres 38738 and eqvrel1cossinidres 38772. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴)
 
Theorempetxrnidres2 38803 Class 𝐴 is a partition by a range Cartesian product with the identity class restricted to it if and only if the cosets by the range Cartesian product are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
(( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( I ↾ 𝐴)) / ≀ (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴))
 
Theorempetxrnidres 38804 A class is a partition by a range Cartesian product with the identity class restricted to it if and only if the cosets by the range Cartesian product are in equivalence relation on it. Cf. br1cossxrnidres 38432, disjALTVxrnidres 38739 and eqvrel1cossxrnidres 38773. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ErALTV 𝐴)
 
Theoremeqvreldisj1 38805* The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj2 38806, eqvreldisj3 38807). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 3-Dec-2024.)
( EqvRel 𝑅 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
 
Theoremeqvreldisj2 38806 The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj3 38807). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 19-Sep-2021.)
( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅))
 
Theoremeqvreldisj3 38807 The elements of the quotient set of an equivalence relation are disjoint (cf. qsdisj2 8833). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 20-Jun-2019.) (Revised by Peter Mazsa, 19-Sep-2021.)
( EqvRel 𝑅 → Disj ( E ↾ (𝐴 / 𝑅)))
 
Theoremeqvreldisj4 38808 Intersection with the converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 31-Dec-2021.)
( EqvRel 𝑅 → Disj (𝑆 ∩ ( E ↾ (𝐵 / 𝑅))))
 
Theoremeqvreldisj5 38809 Range Cartesian product with converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
( EqvRel 𝑅 → Disj (𝑆 ⋉ ( E ↾ (𝐵 / 𝑅))))
 
Theoremeqvrelqseqdisj2 38810 Implication of eqvreldisj2 38806, lemma for The Main Theorem of Equivalences mainer 38815. (Contributed by Peter Mazsa, 23-Sep-2021.)
(( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → ElDisj 𝐴)
 
Theoremfences3 38811 Implication of eqvrelqseqdisj2 38810 and n0eldmqseq 38630, see comment of fences 38825. (Contributed by Peter Mazsa, 30-Dec-2024.)
(( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
 
Theoremeqvrelqseqdisj3 38812 Implication of eqvreldisj3 38807, lemma for the Member Partition Equivalence Theorem mpet3 38817. (Contributed by Peter Mazsa, 27-Oct-2020.) (Revised by Peter Mazsa, 24-Sep-2021.)
(( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj ( E ↾ 𝐴))
 
Theoremeqvrelqseqdisj4 38813 Lemma for petincnvepres2 38829. (Contributed by Peter Mazsa, 31-Dec-2021.)
(( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (𝑆 ∩ ( E ↾ 𝐴)))
 
Theoremeqvrelqseqdisj5 38814 Lemma for the Partition-Equivalence Theorem pet2 38831. (Contributed by Peter Mazsa, 15-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
(( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (𝑆 ⋉ ( E ↾ 𝐴)))
 
Theoremmainer 38815 The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 26-Sep-2021.)
(𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
 
Theorempartimcomember 38816 Partition with general 𝑅 (in addition to the member partition cf. mpet 38820 and mpet2 38821) implies equivalent comembers. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
(𝑅 Part 𝐴 → CoMembEr 𝐴)
 
Theoremmpet3 38817 Member Partition-Equivalence Theorem. Together with mpet 38820 mpet2 38821, mostly in its conventional cpet 38819 and cpet2 38818 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38831 with general 𝑅). (Contributed by Peter Mazsa, 4-May-2018.) (Revised by Peter Mazsa, 26-Sep-2021.)
(( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
 
Theoremcpet2 38818 The conventional form of the Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have called disjoint or partition what we call element disjoint or member partition, see also cpet 38819. Together with cpet 38819, mpet 38820 mpet2 38821, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38831 with general 𝑅). (Contributed by Peter Mazsa, 30-Dec-2024.)
(( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
 
Theoremcpet 38819 The conventional form of Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have been calling disjoint or partition what we call element disjoint or member partition, see also cpet2 38818. Cf. mpet 38820, mpet2 38821 and mpet3 38817 for unconventional forms of Member Partition-Equivalence Theorem. Cf. pet 38832 and pet2 38831 for Partition-Equivalence Theorem with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
( MembPart 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
 
Theoremmpet 38820 Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets 38823. Member partition and comember equivalence relation are the same (or: each element of 𝐴 have equivalent comembers if and only if 𝐴 is a member partition). Together with mpet2 38821, mpet3 38817, and with the conventional cpet 38819 and cpet2 38818, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38831 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.)
( MembPart 𝐴 ↔ CoMembEr 𝐴)
 
Theoremmpet2 38821 Member Partition-Equivalence Theorem in a shorter form. Together with mpet 38820 mpet3 38817, mostly in its conventional cpet 38819 and cpet2 38818 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38831 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.)
(( E ↾ 𝐴) Part 𝐴 ↔ ≀ ( E ↾ 𝐴) ErALTV 𝐴)
 
Theoremmpets2 38822 Member Partition-Equivalence Theorem with binary relations, cf. mpet2 38821. (Contributed by Peter Mazsa, 24-Sep-2021.)
(𝐴𝑉 → (( E ↾ 𝐴) Parts 𝐴 ↔ ≀ ( E ↾ 𝐴) Ers 𝐴))
 
Theoremmpets 38823 Member Partition-Equivalence Theorem in its shortest possible form: it shows that member partitions and comember equivalence relations are literally the same. Cf. pet 38832, the Partition-Equivalence Theorem, with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
MembParts = CoMembErs
 
Theoremmainpart 38824 Partition with general 𝑅 also imply member partition. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
(𝑅 Part 𝐴 → MembPart 𝐴)
 
Theoremfences 38825 The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet 38820) generate a partition of the members. (Contributed by Peter Mazsa, 26-Sep-2021.)
(𝑅 ErALTV 𝐴 → MembPart 𝐴)
 
Theoremfences2 38826 The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet3 38817) generate a partition of the members, it alo means that (𝑅 ErALTV 𝐴 → ElDisj 𝐴) and that (𝑅 ErALTV 𝐴 → ¬ ∅ ∈ 𝐴). (Contributed by Peter Mazsa, 15-Oct-2021.)
(𝑅 ErALTV 𝐴 → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
 
Theoremmainer2 38827 The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 15-Oct-2021.)
(𝑅 ErALTV 𝐴 → ( CoElEqvRel 𝐴 ∧ ¬ ∅ ∈ 𝐴))
 
Theoremmainerim 38828 Every equivalence relation implies equivalent coelements. (Contributed by Peter Mazsa, 20-Oct-2021.)
(𝑅 ErALTV 𝐴 → CoElEqvRel 𝐴)
 
Theorempetincnvepres2 38829 A partition-equivalence theorem with intersection and general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2021.)
(( Disj (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( E ↾ 𝐴)) / (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( E ↾ 𝐴)) / ≀ (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴))
 
Theorempetincnvepres 38830 The shortest form of a partition-equivalence theorem with intersection and general 𝑅. Cf. br1cossincnvepres 38431. Cf. pet 38832. (Contributed by Peter Mazsa, 23-Sep-2021.)
((𝑅 ∩ ( E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( E ↾ 𝐴)) ErALTV 𝐴)
 
Theorempet2 38831 Partition-Equivalence Theorem, with general 𝑅. This theorem (together with pet 38832 and pets 38833) is the main result of my investigation into set theory, see the comment of pet 38832. (Contributed by Peter Mazsa, 24-May-2021.) (Revised by Peter Mazsa, 23-Sep-2021.)
(( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( E ↾ 𝐴)) / (𝑅 ⋉ ( E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( E ↾ 𝐴)) / ≀ (𝑅 ⋉ ( E ↾ 𝐴))) = 𝐴))
 
Theorempet 38832 Partition-Equivalence Theorem with general 𝑅 while preserving the restricted converse epsilon relation of mpet2 38821 (as opposed to petincnvepres 38830). A class is a partition by a range Cartesian product with general 𝑅 and the restricted converse element class if and only if the cosets by the range Cartesian product are in an equivalence relation on it. Cf. br1cossxrncnvepres 38433.

This theorem (together with pets 38833 and pet2 38831) is the main result of my investigation into set theory. It is no more general than the conventional Member Partition-Equivalence Theorem mpet 38820, mpet2 38821 and mpet3 38817 (because you cannot set 𝑅 in this theorem in such a way that you get mpet2 38821), i.e., it is not the hypothetical General Partition-Equivalence Theorem gpet (𝑅 Part 𝐴 ↔ ≀ 𝑅 ErALTV 𝐴), but this one has a general part that mpet2 38821 lacks: 𝑅, which is sufficient for my future application of set theory, for my purpose outside of set theory. (Contributed by Peter Mazsa, 23-Sep-2021.)

((𝑅 ⋉ ( E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ ( E ↾ 𝐴)) ErALTV 𝐴)
 
Theorempets 38833 Partition-Equivalence Theorem with general 𝑅, with binary relations. This theorem (together with pet 38832 and pet2 38831) is the main result of my investigation into set theory, cf. the comment of pet 38832. (Contributed by Peter Mazsa, 23-Sep-2021.)
((𝐴𝑉𝑅𝑊) → ((𝑅 ⋉ ( E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ ( E ↾ 𝐴)) Ers 𝐴))
 
21.27  Mathbox for Rodolfo Medina
 
21.27.1  Partitions
 
Theoremprtlem60 38834 Lemma for prter3 38863. (Contributed by Rodolfo Medina, 9-Oct-2010.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒𝜏)))
 
Theorembicomdd 38835 Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theoremjca2r 38836 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theoremjca3 38837 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜏)       (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏))))
 
Theoremprtlem70 38838 Lemma for prter3 38863: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
((((𝜓𝜂) ∧ ((𝜑𝜃) ∧ (𝜒𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃𝜏)))) ∧ 𝜂))
 
Theoremibdr 38839 Reverse of ibd 269. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(𝜑 → (𝜒 → (𝜓𝜒)))       (𝜑 → (𝜒𝜓))
 
Theoremprtlem100 38840 Lemma for prter3 38863. (Contributed by Rodolfo Medina, 19-Oct-2010.)
(∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))
 
Theoremprtlem5 38841* Lemma for prter1 38860, prter2 38862, prter3 38863 and prtex 38861. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))
 
Theoremprtlem80 38842 Lemma for prter2 38862. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
 
Theorembrabsb2 38843* A closed form of brabsb 5540. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
 
Theoremeqbrrdv2 38844* Other version of eqbrrdiv 5806. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝑥𝐵𝑦))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
 
Theoremprtlem9 38845* Lemma for prter3 38863. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )
 
Theoremprtlem10 38846* Lemma for prter3 38863. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
( Er 𝐴 → (𝑧𝐴 → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ))))
 
Theoremprtlem11 38847 Lemma for prter2 38862. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))
 
Theoremprtlem12 38848* Lemma for prtex 38861 and prter3 38863. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → Rel )
 
Theoremprtlem13 38849* Lemma for prter1 38860, prter2 38862, prter3 38863 and prtex 38861. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
 
Theoremprtlem16 38850* Lemma for prtex 38861, prter2 38862 and prter3 38863. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       dom = 𝐴
 
Theoremprtlem400 38851* Lemma for prter2 38862 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}        ¬ ∅ ∈ ( 𝐴 / )
 
Syntaxwprt 38852 Extend the definition of a wff to include the partition predicate.
wff Prt 𝐴
 
Definitiondf-prt 38853* Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
 
Theoremerprt 38854 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( Er 𝑋 → Prt (𝐴 / ))
 
Theoremprtlem14 38855* Lemma for prter1 38860, prter2 38862 and prtex 38861. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)))
 
Theoremprtlem15 38856* Lemma for prter1 38860 and prtex 38861. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑧𝐴 (𝑢𝑧𝑣𝑧)))
 
Theoremprtlem17 38857* Lemma for prter2 38862. (Contributed by Rodolfo Medina, 15-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))
 
Theoremprtlem18 38858* Lemma for prter2 38862. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
 
Theoremprtlem19 38859* Lemma for prter2 38862. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))
 
Theoremprter1 38860* Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 Er 𝐴)
 
Theoremprtex 38861* The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))
 
Theoremprter2 38862* The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ( 𝐴 / ) = (𝐴 ∖ {∅}))
 
Theoremprter3 38863* For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → = 𝑆)
 
21.28  Mathbox for Norm Megill

We are sad to report the passing of Metamath creator and long-time contributor Norm Megill (1950 - 2021).

Norm of course was the author of the Metamath proof language, the specification, all of the early tools (and some of the later ones), and the foundational work in logic and set theory for set.mm.

His tools, now at https://github.com/metamath/metamath-exe, include a proof verifier, a proof assistant, a proof minimizer, style checking and reformatting, and tools for searching and displaying proofs.

One of his key insights was that formal proofs can exist not only to be verified by computers, but also to be read by humans. Both the specification of the proof format (which stores full proofs, as opposed to the proof templates used by most proof assistants) and the generated web display of Metamath proofs, one of its distinctive features, contribute to this double objective.

Metamath innovated both by using a very simple substitution rule (and then using that to build more complicated notions like free and bound variables) and also by taking the axiom schemas found in many theories and taking them to the next level - by making all axioms, theorems and proofs operate in terms of schemas.

Not content to create Metamath for his own amusement, he also published it for the world and encouraged the development of a community of people who contributed to it and created their own tools. He was an active participant in the Metamath mailing list and other forums until days before his passing.

It is often our custom to supply a quote from someone memorialized in a mathbox entry. And it is difficult to select a quote for someone who has written so much about Metamath over the years. But here is one quote from the Metamath web page which illustrates not just his clear thinking about what Metamath can and cannot do but also his desire to encourage students at all levels:

Q: Will Metamath help me learn abstract mathematics? A: Yes, but probably not by itself. In order to follow a proof in an advanced math textbook, you may need to know prerequisites that could take years to learn. Some people find this frustrating. In contrast, Metamath uses a single, simple substitution rule that allows you to follow any proof mechanically. You can actually jump in anywhere and be convinced that the symbol string you see in a proof step is a consequence of the symbol strings in the earlier steps that it references, even if you don't understand what the symbols mean. But this is quite different from understanding the meaning of the math that results. Metamath alone probably will not give you an intuitive feel for abstract math, in the same way it can be hard to grasp a large computer program just by reading its source code, even though you may understand each individual instruction. However, the Bibliographic Cross-Reference lets you compare informal proofs in math textbooks and see all the implicit missing details "left to the reader."

 
21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16

These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems axc4 , sp 2180, axc7 2315, axc10 2387, axc11 2432, axc11n 2428, axc15 2424, axc9 2384, axc14 2465, and axc16 2258.

 
Axiomax-c5 38864 Axiom of Specialization. A universally quantified wff implies the wff without the universal quantifier (i.e., an instance, or special case, of the generalized wff). In other words, if something is true for all 𝑥, then it is true for any specific 𝑥 (that would typically occur as a free variable in the wff substituted for 𝜑). (A free variable is one that does not occur in the scope of a quantifier: 𝑥 and 𝑦 are both free in 𝑥 = 𝑦, but only 𝑥 is free in 𝑦𝑥 = 𝑦.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1791. Conditional forms of the converse are given by ax-13 2374, ax-c14 38872, ax-c16 38873, and ax-5 1907.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. In our axiomatization, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution (see stdpc4 2065).

An interesting alternate axiomatization uses axc5c711 38899 and ax-c4 38865 in place of ax-c5 38864, ax-4 1805, ax-10 2138, and ax-11 2154.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2180. (Contributed by NM, 3-Jan-1993.) Use sp 2180 instead. (New usage is discouraged.)

(∀𝑥𝜑𝜑)
 
Axiomax-c4 38865 Axiom of Quantified Implication. This axiom moves a universal quantifier from outside to inside an implication, quantifying 𝜓. Notice that 𝑥 must not be a free variable in the antecedent of the quantified implication, and we express this by binding 𝜑 to "protect" the axiom from a 𝜑 containing a free 𝑥. Axiom scheme C4' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc4 2319. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

(∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 
Axiomax-c7 38866 Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use axc5c711 38899 in place of ax-c5 38864, ax-c7 38866, and ax-11 2154.

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc7 2315. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
 
Axiomax-c10 38867 A variant of ax6 2386. Axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc10 2387. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 
Axiomax-c11 38868 Axiom ax-c11 38868 was the original version of ax-c11n 38869 ("n" for "new"), before it was discovered (in May 2008) that the shorter ax-c11n 38869 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc11 2432. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
 
Axiomax-c11n 38869 Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-c11 38868 and was replaced with this shorter ax-c11n 38869 ("n" for "new") in May 2008. The old axiom is proved from this one as Theorem axc11 2432. Conversely, this axiom is proved from ax-c11 38868 as Theorem axc11nfromc11 38907.

This axiom was proved redundant in July 2015. See Theorem axc11n 2428.

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc11n 2428. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Axiomax-c15 38870 Axiom ax-c15 38870 was the original version of ax-12 2174, before it was discovered (in Jan. 2007) that the shorter ax-12 2174 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of "¬ ∀𝑥𝑥 = 𝑦..." as informally meaning "if 𝑥 and 𝑦 are distinct variables then..." The antecedent becomes false if the same variable is substituted for 𝑥 and 𝑦, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form ¬ ∀𝑥𝑥 = 𝑦 a "distinctor".

Interestingly, if the wff expression substituted for 𝜑 contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of Axiom ax-c15 38870 (from which the ax-12 2174 instance follows by Theorem ax12 2425.) The proof is by induction on formula length, using ax12eq 38922 and ax12el 38923 for the basis steps and ax12indn 38924, ax12indi 38925, and ax12inda 38929 for the induction steps. (This paragraph is true provided we use ax-c11 38868 in place of ax-c11n 38869.)

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc15 2424, which should be used instead. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
 
Axiomax-c9 38871 Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever 𝑧 is distinct from 𝑥 and 𝑦, and 𝑥 = 𝑦 is true, then 𝑥 = 𝑦 quantified with 𝑧 is also true. In other words, 𝑧 is irrelevant to the truth of 𝑥 = 𝑦. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc9 2384. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
 
Axiomax-c14 38872 Axiom of Quantifier Introduction. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint). It is redundant if we include ax-5 1907; see Theorem axc14 2465. Alternately, ax-5 1907 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1907. We retain ax-c14 38872 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1907, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc14 2465. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
 
Axiomax-c16 38873* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 1907 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru 5446), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-5 1907; see Theorem axc16 2258. Alternately, ax-5 1907 becomes logically redundant in the presence of this axiom, but without ax-5 1907 we lose the more powerful metalogic that results from being able to express the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). We retain ax-c16 38873 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1907, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc16 2258. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 
21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old

Theorems ax12fromc15 38886 and ax13fromc9 38887 require some intermediate theorems that are included in this section.

 
Theoremaxc5 38874 This theorem repeats sp 2180 under the name axc5 38874, so that the Metamath program "MM> VERIFY MARKUP" command will check that it matches axiom scheme ax-c5 38864. (Contributed by NM, 18-Aug-2017.) (Proof modification is discouraged.) Use sp 2180 instead. (New usage is discouraged.)
(∀𝑥𝜑𝜑)
 
Theoremax4fromc4 38875 Rederivation of Axiom ax-4 1805 from ax-c4 38865, ax-c5 38864, ax-gen 1791 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2319 for the derivation of ax-c4 38865 from ax-4 1805. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-4 1805 instead. (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 
Theoremax10fromc7 38876 Rederivation of Axiom ax-10 2138 from ax-c7 38866, ax-c4 38865, ax-c5 38864, ax-gen 1791 and propositional calculus. See axc7 2315 for the derivation of ax-c7 38866 from ax-10 2138. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-10 2138 instead. (New usage is discouraged.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremax6fromc10 38877 Rederivation of Axiom ax-6 1964 from ax-c7 38866, ax-c10 38867, ax-gen 1791 and propositional calculus. See axc10 2387 for the derivation of ax-c10 38867 from ax-6 1964. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) Use ax-6 1964 instead. (New usage is discouraged.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
Theoremhba1-o 38878 The setvar 𝑥 is not free in 𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theoremaxc4i-o 38879 Inference version of ax-c4 38865. (Contributed by NM, 3-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)
 
Theoremequid1 38880 Proof of equid 2008 from our older axioms. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1907; see the proof of equid 2008. See equid1ALT 38906 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥
 
Theoremequcomi1 38881 Proof of equcomi 2013 from equid1 38880, avoiding use of ax-5 1907 (the only use of ax-5 1907 is via ax7 2012, so using ax-7 2004 instead would remove dependency on ax-5 1907). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦𝑦 = 𝑥)
 
Theoremaecom-o 38882 Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2429 using ax-c11 38868. Unlike axc11nfromc11 38907, this version does not require ax-5 1907 (see comment of equcomi1 38881). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Theoremaecoms-o 38883 A commutation rule for identical variable specifiers. Version of aecoms 2430 using ax-c11 38868. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)
 
Theoremhbae-o 38884 All variables are effectively bound in an identical variable specifier. Version of hbae 2433 using ax-c11 38868. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
 
Theoremdral1-o 38885 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2441 using ax-c11 38868. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 
Theoremax12fromc15 38886 Rederivation of Axiom ax-12 2174 from ax-c15 38870, ax-c11 38868 (used through dral1-o 38885), and other older axioms. See Theorem axc15 2424 for the derivation of ax-c15 38870 from ax-12 2174.

An open problem is whether we can prove this using ax-c11n 38869 instead of ax-c11 38868.

This proof uses newer axioms ax-4 1805 and ax-6 1964, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 38865 and ax-c10 38867. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremax13fromc9 38887 Derive ax-13 2374 from ax-c9 38871 and other older axioms.

This proof uses newer axioms ax-4 1805 and ax-6 1964, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 38865 and ax-c10 38867. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
21.28.3  Legacy theorems using obsolete axioms

These theorems were mostly intended to study properties of the older axiom schemes and are not useful outside of this section. They should not be used outside of this section. They may be deleted when they are deemed to no longer be of interest.

 
Theoremax5ALT 38888* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(This theorem simply repeats ax-5 1907 so that we can include the following note, which applies only to the obsolete axiomatization.)

This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1791, ax-c4 38865, ax-c5 38864, ax-11 2154, ax-c7 38866, ax-7 2004, ax-c9 38871, ax-c10 38867, ax-c11 38868, ax-8 2107, ax-9 2115, ax-c14 38872, ax-c15 38870, and ax-c16 38873: in that system, we can derive any instance of ax-5 1907 not containing wff variables by induction on formula length, using ax5eq 38913 and ax5el 38918 for the basis together with hbn 2293, hbal 2164, and hbim 2297. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜑 → ∀𝑥𝜑)
 
Theoremsps-o 38889 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)
 
Theoremhbequid 38890 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof does not use ax-c10 38867.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
 
Theoremnfequid-o 38891 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof uses only ax-4 1805, ax-7 2004, ax-c9 38871, and ax-gen 1791. This shows that this can be proved without ax6 2386, even though Theorem equid 2008 cannot. A shorter proof using ax6 2386 is obtainable from equid 2008 and hbth 1799.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1965, which is used for the derivation of axc9 2384, unless we consider ax-c9 38871 the starting axiom rather than ax-13 2374. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥 = 𝑥
 
Theoremaxc5c7 38892 Proof of a single axiom that can replace ax-c5 38864 and ax-c7 38866. See axc5c7toc5 38893 and axc5c7toc7 38894 for the rederivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)
 
Theoremaxc5c7toc5 38893 Rederivation of ax-c5 38864 from axc5c7 38892. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)
 
Theoremaxc5c7toc7 38894 Rederivation of ax-c7 38866 from axc5c7 38892. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
 
Theoremaxc711 38895 Proof of a single axiom that can replace both ax-c7 38866 and ax-11 2154. See axc711toc7 38897 and axc711to11 38898 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ∀𝑦𝜑)
 
Theoremnfa1-o 38896 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑
 
Theoremaxc711toc7 38897 Rederivation of ax-c7 38866 from axc711 38895. Note that ax-c7 38866 and ax-11 2154 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
 
Theoremaxc711to11 38898 Rederivation of ax-11 2154 from axc711 38895. Note that ax-c7 38866 and ax-11 2154 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremaxc5c711 38899 Proof of a single axiom that can replace ax-c5 38864, ax-c7 38866, and ax-11 2154 in a subsystem that includes these axioms plus ax-c4 38865 and ax-gen 1791 (and propositional calculus). See axc5c711toc5 38900, axc5c711toc7 38901, and axc5c711to11 38902 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 38892. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
 
Theoremaxc5c711toc5 38900 Rederivation of ax-c5 38864 from axc5c711 38899. Only propositional calculus is used by the rederivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49035
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