Theorem prex 2857

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 2517), 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 2509	. . . . 5 |- (x = A -> {x, B} = {A, B})
2	1	eleq1d 1583	. . . 4 |- (x = A -> ({x, B} e. V <-> {A, B} e. V))
3		preq2 2510	. . . . . 6 |- (y = B -> {x, y} = {x, B})
4	3	eleq1d 1583	. . . . 5 |- (y = B -> ({x, y} e. V <-> {x, B} e. V))
5		zfpair2 2856	. . . . 5 |- {x, y} e. V
6	4, 5	vtoclg 1893	. . . 4 |- (B e. V -> {x, B} e. V)
7	2, 6	syl5bi 206	. . 3 |- (x = A -> (B e. V -> {A, B} e. V))
8	7	vtocleg 1901	. 2 |- (A e. V -> (B e. V -> {A, B} e. V))
9		prprc1 2515	. . 3 |- (-. A e. V -> {A, B} = {B})
10		snex 2826	. . 3 |- {B} e. V
11	9, 10	syl6eqel 1599	. 2 |- (-. A e. V -> {A, B} e. V)
12		prprc2 2516	. . 3 |- (-. B e. V -> {A, B} = {A})
13		snex 2826	. . 3 |- {A} e. V
14	12, 13	syl6eqel 1599	. 2 |- (-. B e. V -> {A, B} e. V)
15	8, 11, 14	pm2.61nii 129	1 |- {A, B} e. V

Syntax hints:  -. wn 2   -> wi 3   = wceq 992   e. wcel 994  Vcvv 1857  {csn 2467  {cpr 2468

This theorem is referenced by:  opex 2858  opi2 2861  opth 2863  opeqsn 2879  opeqpr 2880  opthwiener 2884  uniop 2885  unex 3095  tpex 3102  op1stb 3136  xpsspw 3346  relop 3365  opthreg 4749  rankop 4839  aceq6b 4888  xrex 5646  unctb 7789  indistop 7860  spwpr4 8925  spwpr4OLD 8926  spwpr4aOLD 8927  unpde2eg22 10826  set2elt 10827  cnfilca 11088

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471

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