Theorem zfpair2 5390

Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 5389. See zfpair 5377 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 5389	. . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
2	1	sepexi 5250	. . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
3		dfcleq 2754	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
4		vex 3457	. . . . . . . 8 ⊢ 𝑤 ∈ V
5	4	elpr 4606	. . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
6	5	bibi2i 339	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
7	6	albii 1838	. . . . 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 1867	. . 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 3472	1 ⊢ {𝑥, 𝑦} ∈ V

Syntax hints:   ↔ wb 208   ∨ wo 858  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453  {cpr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-sn 4582  df-pr 4584

This theorem is referenced by:  vsnex  5391  prexOLD  5399  pwssun  5537  xpsspw  5780  funopg  6551  fiint  9267  brdom7disj  10485  brdom6disj  10486  2pthfrgrrn  30430  sprval  48049  prprval  48084  reupr  48092