Theorem bj-zfpair2 16505

Description: Proof of zfpair2 4300 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 16415	. . . . 5 ⊢ BOUNDED 𝑤 = 𝑥
2		ax-bdeq 16415	. . . . 5 ⊢ BOUNDED 𝑤 = 𝑦
3	1, 2	ax-bdor 16411	. . . 4 ⊢ BOUNDED (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)
4		ax-pr 4299	. . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
5	3, 4	bdbm1.3ii 16486	. . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
6		dfcleq 2225	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
7		vex 2805	. . . . . . . 8 ⊢ 𝑤 ∈ V
8	7	elpr 3690	. . . . . . 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 1518	. . . . 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 1653	. . 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 2812	1 ⊢ {𝑥, 𝑦} ∈ V

Syntax hints:   ↔ wb 105   ∨ wo 715  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802  {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-pr 4299  ax-bdor 16411  ax-bdeq 16415  ax-bdsep 16479

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  bj-prexg  16506