Theorem prex 2776

Description: The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51. By virtue of its definition, an unordered pair remains a set (even though no longer a pair) even when its components are proper classes (see prprc 2450), so we can dispense with hypotheses requiring them to be sets.

Assertion

Ref	Expression
prex	|- {A, B} e. V

Proof of Theorem prex

Step	Hyp	Ref	Expression
1		preq1 2444	. . . . 5 |- (x = A -> {x, B} = {A, B})
2	1	eleq1d 1537	. . . 4 |- (x = A -> ({x, B} e. V <-> {A, B} e. V))
3		preq2 2445	. . . . . 6 |- (y = B -> {x, y} = {x, B})
4	3	eleq1d 1537	. . . . 5 |- (y = B -> ({x, y} e. V <-> {x, B} e. V))
5		zfpair2 2775	. . . . 5 |- {x, y} e. V
6	4, 5	vtoclg 1843	. . . 4 |- (B e. V -> {x, B} e. V)
7	2, 6	syl5bi 208	. . 3 |- (x = A -> (B e. V -> {A, B} e. V))
8	7	vtocleg 1851	. 2 |- (A e. V -> (B e. V -> {A, B} e. V))
9		prprc1 2448	. . 3 |- (-. A e. V -> {A, B} = {B})
10		snex 2745	. . 3 |- {B} e. V
11	9, 10	syl6eqel 1553	. 2 |- (-. A e. V -> {A, B} e. V)
12		prprc2 2449	. . 3 |- (-. B e. V -> {A, B} = {A})
13		snex 2745	. . 3 |- {A} e. V
14	12, 13	syl6eqel 1553	. 2 |- (-. B e. V -> {A, B} e. V)
15	8, 11, 14	pm2.61nii 131	1 |- {A, B} e. V

Syntax hints:  -. wn 2   -> wi 3   = wceq 954   e. wcel 956  Vcvv 1807  {csn 2405  {cpr 2406

This theorem is referenced by:  opex 2777  opi2 2780  opth 2782  opeqsn 2797  opeqpr 2798  opthwiener 2802  uniop 2803  unex 2867  tpex 2873  op1stb 2908  xpsspw 3252  relop 3270  opthreg 4584  rankop 4673  aceq6b 4722  xrex 5472  unctb 7527  indistop 7598  cnfilca 10487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409

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