Theorem prex 2781

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 2454), 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 2448	. . . . 5 |- (x = A -> {x, B} = {A, B})
2	1	eleq1d 1540	. . . 4 |- (x = A -> ({x, B} e. V <-> {A, B} e. V))
3		preq2 2449	. . . . . 6 |- (y = B -> {x, y} = {x, B})
4	3	eleq1d 1540	. . . . 5 |- (y = B -> ({x, y} e. V <-> {x, B} e. V))
5		zfpair2 2780	. . . . 5 |- {x, y} e. V
6	4, 5	vtoclg 1847	. . . 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 1855	. 2 |- (A e. V -> (B e. V -> {A, B} e. V))
9		prprc1 2452	. . 3 |- (-. A e. V -> {A, B} = {B})
10		snex 2750	. . 3 |- {B} e. V
11	9, 10	syl6eqel 1556	. 2 |- (-. A e. V -> {A, B} e. V)
12		prprc2 2453	. . 3 |- (-. B e. V -> {A, B} = {A})
13		snex 2750	. . 3 |- {A} e. V
14	12, 13	syl6eqel 1556	. 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 956   e. wcel 958  Vcvv 1811  {csn 2409  {cpr 2410

This theorem is referenced by:  opex 2782  opi2 2785  opth 2787  opeqsn 2802  opeqpr 2803  opthwiener 2807  uniop 2808  unex 2872  tpex 2878  op1stb 2913  xpsspw 3257  relop 3275  opthreg 4604  rankop 4693  aceq6b 4742  xrex 5492  unctb 7577  indistop 7648  spwpr4OLD 8663  spwpr4aOLD 8664  set2elt 10545  cnfilca 10583  cnfilcaOLD 10584

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413

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