Theorem bj-zfpair2 15015

Description: Proof of zfpair2 4222 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 14925	. . . . 5 ⊢ BOUNDED 𝑤 = 𝑥
2		ax-bdeq 14925	. . . . 5 ⊢ BOUNDED 𝑤 = 𝑦
3	1, 2	ax-bdor 14921	. . . 4 ⊢ BOUNDED (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)
4		ax-pr 4221	. . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
5	3, 4	bdbm1.3ii 14996	. . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
6		dfcleq 2181	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
7		vex 2752	. . . . . . . 8 ⊢ 𝑤 ∈ V
8	7	elpr 3625	. . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
9	8	bibi2i 227	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
10	9	albii 1480	. . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
11	6, 10	bitri 184	. . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
12	11	exbii 1615	. . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
13	5, 12	mpbir 146	. 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦}
14	13	issetri 2758	1 ⊢ {𝑥, 𝑦} ∈ V

Syntax hints:   ↔ wb 105   ∨ wo 709  ∀wal 1361   = wceq 1363  ∃wex 1502   ∈ wcel 2158  Vcvv 2749  {cpr 3605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-pr 4221  ax-bdor 14921  ax-bdeq 14925  ax-bdsep 14989

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611

This theorem is referenced by:  bj-prexg  15016