Theorem bj-zfpair2 13792

Description: Proof of zfpair2 4188 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 13702	. . . . 5 ⊢ BOUNDED 𝑤 = 𝑥
2		ax-bdeq 13702	. . . . 5 ⊢ BOUNDED 𝑤 = 𝑦
3	1, 2	ax-bdor 13698	. . . 4 ⊢ BOUNDED (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)
4		ax-pr 4187	. . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
5	3, 4	bdbm1.3ii 13773	. . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
6		dfcleq 2159	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
7		vex 2729	. . . . . . . 8 ⊢ 𝑤 ∈ V
8	7	elpr 3597	. . . . . . 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 1458	. . . . 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 1593	. . 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 2735	1 ⊢ {𝑥, 𝑦} ∈ V

Syntax hints:   ↔ wb 104   ∨ wo 698  ∀wal 1341   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726  {cpr 3577

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-pr 4187  ax-bdor 13698  ax-bdeq 13702  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  bj-prexg  13793