Theorem prex 1892

Description: The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51. Note that an unordered pair is a set even if its components are proper classes.

Assertion

Ref	Expression
prex	⊢ {A, B} ∈ V

Proof of Theorem prex

Step	Hyp	Ref	Expression
1		preq1 1870	. . . . . 6 ⊢ (x = A → {x, B} = {A, B})
2	1	eleq1d 1155	. . . . 5 ⊢ (x = A → ({x, B} ∈ V ↔ {A, B} ∈ V))
3		preq2 1871	. . . . . . 7 ⊢ (y = B → {x, y} = {x, B})
4	3	eleq1d 1155	. . . . . 6 ⊢ (y = B → ({x, y} ∈ V ↔ {x, B} ∈ V))
5		zfpair 1891	. . . . . 6 ⊢ {x, y} ∈ V
6	4, 5	vtoclg 1383	. . . . 5 ⊢ (B ∈ V → {x, B} ∈ V)
7	2, 6	syl5bi 183	. . . 4 ⊢ (x = A → (B ∈ V → {A, B} ∈ V))
8	7	vtocleg 1390	. . 3 ⊢ (A ∈ V → (B ∈ V → {A, B} ∈ V))
9	8	imp 277	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V)
10		ianor 253	. . 3 ⊢ (¬ (A ∈ V ∧ B ∈ V) ↔ (¬ A ∈ V ∨ ¬ B ∈ V))
11		prprc 1839	. . . . 5 ⊢ (¬ A ∈ V → {A, B} = {B})
12		snex 1859	. . . . . 6 ⊢ {B} ∈ V
13		eleq1 1149	. . . . . 6 ⊢ ({A, B} = {B} → ({A, B} ∈ V ↔ {B} ∈ V))
14	12, 13	mpbiri 169	. . . . 5 ⊢ ({A, B} = {B} → {A, B} ∈ V)
15	11, 14	syl 12	. . . 4 ⊢ (¬ A ∈ V → {A, B} ∈ V)
16		prprc 1839	. . . . . 6 ⊢ (¬ B ∈ V → {B, A} = {A})
17		prcom 1840	. . . . . 6 ⊢ {A, B} = {B, A}
18	16, 17	syl5eq 1136	. . . . 5 ⊢ (¬ B ∈ V → {A, B} = {A})
19		snex 1859	. . . . . 6 ⊢ {A} ∈ V
20		eleq1 1149	. . . . . 6 ⊢ ({A, B} = {A} → ({A, B} ∈ V ↔ {A} ∈ V))
21	19, 20	mpbiri 169	. . . . 5 ⊢ ({A, B} = {A} → {A, B} ∈ V)
22	18, 21	syl 12	. . . 4 ⊢ (¬ B ∈ V → {A, B} ∈ V)
23	15, 22	jaoi 275	. . 3 ⊢ ((¬ A ∈ V ∨ ¬ B ∈ V) → {A, B} ∈ V)
24	10, 23	sylbi 174	. 2 ⊢ (¬ (A ∈ V ∧ B ∈ V) → {A, B} ∈ V)
25	9, 24	pm2.61i 110	1 ⊢ {A, B} ∈ V

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  {cpr 1809

This theorem is referenced by:  opex 1893  opi2 1896  opth 1898  opthwiener 1914  unop 1931  unex 1949  tpex 1952  op1stb 1992  xpex 2488  opthreg 3455  aceq6b 3565

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812

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