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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | petinidres2 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)) | ||
Theorem | petinidres 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) | ||
Theorem | petxrnidres2 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)) | ||
Theorem | petxrnidres 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) | ||
Theorem | eqvreldisj1 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 ∨ (𝑥 ∩ 𝑦) = ∅)) | ||
Theorem | eqvreldisj2 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)) | ||
Theorem | eqvreldisj3 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))) | ||
Theorem | eqvreldisj4 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)))) | ||
Theorem | eqvreldisj5 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)))) | ||
Theorem | eqvrelqseqdisj2 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) | ||
Theorem | fences3 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)) | ||
Theorem | eqvrelqseqdisj3 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)) | ||
Theorem | eqvrelqseqdisj4 38813 | Lemma for petincnvepres2 38829. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (𝑆 ∩ (◡ E ↾ 𝐴))) | ||
Theorem | eqvrelqseqdisj5 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))) | ||
Theorem | mainer 38815 | The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 26-Sep-2021.) |
⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴) | ||
Theorem | partimcomember 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) | ||
Theorem | mpet3 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)) | ||
Theorem | cpet2 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)) | ||
Theorem | cpet 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)) | ||
Theorem | mpet 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) | ||
Theorem | mpet2 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) | ||
Theorem | mpets2 38822 | Member Partition-Equivalence Theorem with binary relations, cf. mpet2 38821. (Contributed by Peter Mazsa, 24-Sep-2021.) |
⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) Parts 𝐴 ↔ ≀ (◡ E ↾ 𝐴) Ers 𝐴)) | ||
Theorem | mpets 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 | ||
Theorem | mainpart 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) | ||
Theorem | fences 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) | ||
Theorem | fences2 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)) | ||
Theorem | mainer2 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)) | ||
Theorem | mainerim 38828 | Every equivalence relation implies equivalent coelements. (Contributed by Peter Mazsa, 20-Oct-2021.) |
⊢ (𝑅 ErALTV 𝐴 → CoElEqvRel 𝐴) | ||
Theorem | petincnvepres2 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)) | ||
Theorem | petincnvepres 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) | ||
Theorem | pet2 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)) | ||
Theorem | pet 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) | ||
Theorem | pets 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴)) | ||
Theorem | prtlem60 38834 | Lemma for prter3 38863. (Contributed by Rodolfo Medina, 9-Oct-2010.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜓 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) | ||
Theorem | bicomdd 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 → (𝜒 ↔ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜒))) | ||
Theorem | jca2r 38836 | Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜓 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 ∧ 𝜒))) | ||
Theorem | jca3 38837 | Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → 𝜏) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 ∧ 𝜏)))) | ||
Theorem | prtlem70 38838 | Lemma for prter3 38863: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.) |
⊢ ((((𝜓 ∧ 𝜂) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃 ∧ 𝜏)))) ∧ 𝜂)) | ||
Theorem | ibdr 38839 | Reverse of ibd 269. (Contributed by Rodolfo Medina, 30-Sep-2010.) |
⊢ (𝜑 → (𝜒 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||
Theorem | prtlem100 38840 | Lemma for prter3 38863. (Contributed by Rodolfo Medina, 19-Oct-2010.) |
⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑)) | ||
Theorem | prtlem5 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]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥)) | ||
Theorem | prtlem80 38842 | Lemma for prter2 38862. (Contributed by Rodolfo Medina, 17-Oct-2010.) |
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴})) | ||
Theorem | brabsb2 38843* | A closed form of brabsb 5540. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)) | ||
Theorem | eqbrrdv2 38844* | Other version of eqbrrdiv 5806. (Contributed by Rodolfo Medina, 30-Sep-2010.) |
⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) ⇒ ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) | ||
Theorem | prtlem9 38845* | Lemma for prter3 38863. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) | ||
Theorem | prtlem10 38846* | Lemma for prter3 38863. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ )))) | ||
Theorem | prtlem11 38847 | Lemma for prter2 38862. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) | ||
Theorem | prtlem12 38848* | Lemma for prtex 38861 and prter3 38863. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ ( ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ ) | ||
Theorem | prtlem13 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〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) | ||
Theorem | prtlem16 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〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ dom ∼ = ∪ 𝐴 | ||
Theorem | prtlem400 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〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ ) | ||
Syntax | wprt 38852 | Extend the definition of a wff to include the partition predicate. |
wff Prt 𝐴 | ||
Definition | df-prt 38853* | Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | ||
Theorem | erprt 38854 | The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ )) | ||
Theorem | prtlem14 38855* | Lemma for prter1 38860, prter2 38862 and prtex 38861. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦))) | ||
Theorem | prtlem15 38856* | Lemma for prter1 38860 and prtex 38861. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → ∃𝑧 ∈ 𝐴 (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧))) | ||
Theorem | prtlem17 38857* | Lemma for prter2 38862. (Contributed by Rodolfo Medina, 15-Oct-2010.) |
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥))) | ||
Theorem | prtlem18 38858* | Lemma for prter2 38862. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤))) | ||
Theorem | prtlem19 38859* | Lemma for prter2 38862. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ )) | ||
Theorem | prter1 38860* | Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴) | ||
Theorem | prtex 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〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ 𝐴 ∈ V)) | ||
Theorem | prter2 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〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → (∪ 𝐴 / ∼ ) = (𝐴 ∖ {∅})) | ||
Theorem | prter3 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〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → ∼ = 𝑆) | ||
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." | ||
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. | ||
Axiom | ax-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.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Axiom | ax-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(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Axiom | ax-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 ¬ ∀𝑥𝜑 → 𝜑) | ||
Axiom | ax-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) | ||
Axiom | ax-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)) | ||
Axiom | ax-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) | ||
Axiom | ax-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 = 𝑦 → 𝜑)))) | ||
Axiom | ax-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))) | ||
Axiom | ax-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 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))) | ||
Axiom | ax-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)) | ||
Theorems ax12fromc15 38886 and ax13fromc9 38887 require some intermediate theorems that are included in this section. | ||
Theorem | axc5 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.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | ax4fromc4 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(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | ax10fromc7 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) | ||
Theorem | ax6fromc10 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 | ||
Theorem | hba1-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 → ∀𝑥∀𝑥𝜑) | ||
Theorem | axc4i-o 38879 | Inference version of ax-c4 38865. (Contributed by NM, 3-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | equid1 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 | ||
Theorem | equcomi1 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 = 𝑦 → 𝑦 = 𝑥) | ||
Theorem | aecom-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) | ||
Theorem | aecoms-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 = 𝑦 → 𝜑) ⇒ ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | hbae-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 → ∀𝑧∀𝑥 𝑥 = 𝑦) | ||
Theorem | dral1-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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
Theorem | ax12fromc15 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 = 𝑦 → 𝜑))) | ||
Theorem | ax13fromc9 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)) | ||
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. | ||
Theorem | ax5ALT 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) | ||
Theorem | sps-o 38889 | Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | hbequid 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) | ||
Theorem | nfequid-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.) |
⊢ Ⅎ𝑦 𝑥 = 𝑥 | ||
Theorem | axc5c7 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) | ||
Theorem | axc5c7toc5 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.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | axc5c7toc7 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 ¬ ∀𝑥𝜑 → 𝜑) | ||
Theorem | axc711 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 ¬ ∀𝑦∀𝑥𝜑 → ∀𝑦𝜑) | ||
Theorem | nfa1-o 38896 | 𝑥 is not free in ∀𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥∀𝑥𝜑 | ||
Theorem | axc711toc7 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 ¬ ∀𝑥𝜑 → 𝜑) | ||
Theorem | axc711to11 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.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | axc5c711 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.) |
⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | axc5c711toc5 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.) |
⊢ (∀𝑥𝜑 → 𝜑) |
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