Theorem bj-zfpair2 6812

Description: Proof of zfpair2 3917 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-zfpair2	⊢ {x, y} ∈ V

Proof of Theorem bj-zfpair2

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

Step	Hyp	Ref	Expression
1		ax-bdeq 6726	. . . . 5 ⊢ BOUNDED w = x
2		ax-bdeq 6726	. . . . 5 ⊢ BOUNDED w = y
3	1, 2	ax-bdor 6722	. . . 4 ⊢ BOUNDED (w = x ∨ w = y)
4		ax-pr 3916	. . . 4 ⊢ ∃z∀w((w = x ∨ w = y) → w ∈ z)
5	3, 4	bdbm1.3ii 6796	. . 3 ⊢ ∃z∀w(w ∈ z ↔ (w = x ∨ w = y))
6		dfcleq 2016	. . . . 5 ⊢ (z = {x, y} ↔ ∀w(w ∈ z ↔ w ∈ {x, y}))
7		vex 2536	. . . . . . . 8 ⊢ w ∈ V
8	7	elpr 3366	. . . . . . 7 ⊢ (w ∈ {x, y} ↔ (w = x ∨ w = y))
9	8	bibi2i 216	. . . . . 6 ⊢ ((w ∈ z ↔ w ∈ {x, y}) ↔ (w ∈ z ↔ (w = x ∨ w = y)))
10	9	albii 1339	. . . . 5 ⊢ (∀w(w ∈ z ↔ w ∈ {x, y}) ↔ ∀w(w ∈ z ↔ (w = x ∨ w = y)))
11	6, 10	bitri 173	. . . 4 ⊢ (z = {x, y} ↔ ∀w(w ∈ z ↔ (w = x ∨ w = y)))
12	11	exbii 1478	. . 3 ⊢ (∃z z = {x, y} ↔ ∃z∀w(w ∈ z ↔ (w = x ∨ w = y)))
13	5, 12	mpbir 134	. 2 ⊢ ∃z z = {x, y}
14	13	issetri 2540	1 ⊢ {x, y} ∈ V

Syntax hints:   ↔ wb 98   ∨ wo 616  ∀wal 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2533  {cpr 3349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-pr 3916  ax-bdor 6722  ax-bdeq 6726  ax-bdsep 6790

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-un 2897  df-sn 3354  df-pr 3355

This theorem is referenced by:  bj-prexg  6813