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Theorem brabvv 5823
Description: If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.)
Assertion
Ref Expression
brabvv (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Distinct variable groups:   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem brabvv
StepHypRef Expression
1 df-br 3936 . . . . . 6 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2 elopab 4186 . . . . . 6 (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
31, 2bitri 183 . . . . 5 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 exsimpl 1597 . . . . . 6 (∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
54eximi 1580 . . . . 5 (∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
63, 5sylbi 120 . . . 4 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → ∃𝑥𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
7 vex 2692 . . . . . . . 8 𝑥 ∈ V
8 vex 2692 . . . . . . . 8 𝑦 ∈ V
97, 8opth 4165 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
109biimpi 119 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ → (𝑥 = 𝑋𝑦 = 𝑌))
1110eqcoms 2143 . . . . 5 (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝑋𝑦 = 𝑌))
12112eximi 1581 . . . 4 (∃𝑥𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦(𝑥 = 𝑋𝑦 = 𝑌))
136, 12syl 14 . . 3 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → ∃𝑥𝑦(𝑥 = 𝑋𝑦 = 𝑌))
14 eeanv 1905 . . 3 (∃𝑥𝑦(𝑥 = 𝑋𝑦 = 𝑌) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
1513, 14sylib 121 . 2 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
16 isset 2695 . . 3 (𝑋 ∈ V ↔ ∃𝑥 𝑥 = 𝑋)
17 isset 2695 . . 3 (𝑌 ∈ V ↔ ∃𝑦 𝑦 = 𝑌)
1816, 17anbi12i 456 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
1915, 18sylibr 133 1 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wex 1469  wcel 1481  Vcvv 2689  cop 3533   class class class wbr 3935  {copab 3994
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-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  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 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-br 3936  df-opab 3996
This theorem is referenced by: (None)
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