Theorem bj-zfpair2 13108

Description: Proof of zfpair2 4132 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 13018	. . . . 5 ⊢ BOUNDED 𝑤 = 𝑥
2		ax-bdeq 13018	. . . . 5 ⊢ BOUNDED 𝑤 = 𝑦
3	1, 2	ax-bdor 13014	. . . 4 ⊢ BOUNDED (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)
4		ax-pr 4131	. . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
5	3, 4	bdbm1.3ii 13089	. . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
6		dfcleq 2133	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
7		vex 2689	. . . . . . . 8 ⊢ 𝑤 ∈ V
8	7	elpr 3548	. . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
9	8	bibi2i 226	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
10	9	albii 1446	. . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
11	6, 10	bitri 183	. . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
12	11	exbii 1584	. . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
13	5, 12	mpbir 145	. 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦}
14	13	issetri 2695	1 ⊢ {𝑥, 𝑦} ∈ V

Syntax hints:   ↔ wb 104   ∨ wo 697  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  {cpr 3528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-pr 4131  ax-bdor 13014  ax-bdeq 13018  ax-bdsep 13082

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534

This theorem is referenced by:  bj-prexg  13109