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Theorem bj-zfpair2 13277
Description: Proof of zfpair2 4139 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 13187 . . . . 5 BOUNDED 𝑤 = 𝑥
2 ax-bdeq 13187 . . . . 5 BOUNDED 𝑤 = 𝑦
31, 2ax-bdor 13183 . . . 4 BOUNDED (𝑤 = 𝑥𝑤 = 𝑦)
4 ax-pr 4138 . . . 4 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
53, 4bdbm1.3ii 13258 . . 3 𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦))
6 dfcleq 2134 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
7 vex 2692 . . . . . . . 8 𝑤 ∈ V
87elpr 3552 . . . . . . 7 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
98bibi2i 226 . . . . . 6 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
109albii 1447 . . . . 5 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
116, 10bitri 183 . . . 4 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
1211exbii 1585 . . 3 (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
135, 12mpbir 145 . 2 𝑧 𝑧 = {𝑥, 𝑦}
1413issetri 2698 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff set class
Syntax hints:  wb 104  wo 698  wal 1330   = wceq 1332  wex 1469  wcel 1481  Vcvv 2689  {cpr 3532
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-pr 4138  ax-bdor 13183  ax-bdeq 13187  ax-bdsep 13251
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538
This theorem is referenced by:  bj-prexg  13278
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