Theorem zfpair2 5322

Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 5321. See zfpair 5313 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)

Assertion

Ref	Expression
zfpair2	⊢ {𝑥, 𝑦} ∈ V

Proof of Theorem zfpair2

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pr 5321	. . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
2	1	bm1.3ii 5198	. . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
3		dfcleq 2815	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
4		vex 3497	. . . . . . . 8 ⊢ 𝑤 ∈ V
5	4	elpr 4583	. . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
6	5	bibi2i 340	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
7	6	albii 1816	. . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
8	3, 7	bitri 277	. . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
9	8	exbii 1844	. . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
10	2, 9	mpbir 233	. 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦}
11	10	issetri 3510	1 ⊢ {𝑥, 𝑦} ∈ V

Syntax hints:   ↔ wb 208   ∨ wo 843  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3494  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3940  df-sn 4561  df-pr 4563

This theorem is referenced by:  snex  5323  prex  5324  pwssun  5449  xpsspw  5676  funopg  6383  fiint  8789  brdom7disj  9947  brdom6disj  9948  2pthfrgrrn  28055  sprval  43635  prprval  43670  reupr  43678