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Theorem bj-zfpair2 14933
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.
StepHypRef Expression
1 ax-bdeq 14843 . . . . 5 BOUNDED 𝑤 = 𝑥
2 ax-bdeq 14843 . . . . 5 BOUNDED 𝑤 = 𝑦
31, 2ax-bdor 14839 . . . 4 BOUNDED (𝑤 = 𝑥𝑤 = 𝑦)
4 ax-pr 4221 . . . 4 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
53, 4bdbm1.3ii 14914 . . 3 𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦))
6 dfcleq 2181 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
7 vex 2752 . . . . . . . 8 𝑤 ∈ V
87elpr 3625 . . . . . . 7 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
98bibi2i 227 . . . . . 6 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
109albii 1480 . . . . 5 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
116, 10bitri 184 . . . 4 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
1211exbii 1615 . . 3 (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
135, 12mpbir 146 . 2 𝑧 𝑧 = {𝑥, 𝑦}
1413issetri 2758 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff set class
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 14839  ax-bdeq 14843  ax-bdsep 14907
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  14934
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