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Theorem brabvv 5453
 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 3718 . . . . . 6 (𝑋{⟨x, y⟩ ∣ φ}𝑌 ↔ ⟨𝑋, 𝑌 {⟨x, y⟩ ∣ φ})
2 elopab 3948 . . . . . 6 (⟨𝑋, 𝑌 {⟨x, y⟩ ∣ φ} ↔ xy(⟨𝑋, 𝑌⟩ = ⟨x, y φ))
31, 2bitri 173 . . . . 5 (𝑋{⟨x, y⟩ ∣ φ}𝑌xy(⟨𝑋, 𝑌⟩ = ⟨x, y φ))
4 exsimpl 1492 . . . . . 6 (y(⟨𝑋, 𝑌⟩ = ⟨x, y φ) → y𝑋, 𝑌⟩ = ⟨x, y⟩)
54eximi 1475 . . . . 5 (xy(⟨𝑋, 𝑌⟩ = ⟨x, y φ) → xy𝑋, 𝑌⟩ = ⟨x, y⟩)
63, 5sylbi 114 . . . 4 (𝑋{⟨x, y⟩ ∣ φ}𝑌xy𝑋, 𝑌⟩ = ⟨x, y⟩)
7 vex 2537 . . . . . . . 8 x V
8 vex 2537 . . . . . . . 8 y V
97, 8opth 3927 . . . . . . 7 (⟨x, y⟩ = ⟨𝑋, 𝑌⟩ ↔ (x = 𝑋 y = 𝑌))
109biimpi 113 . . . . . 6 (⟨x, y⟩ = ⟨𝑋, 𝑌⟩ → (x = 𝑋 y = 𝑌))
1110eqcoms 2026 . . . . 5 (⟨𝑋, 𝑌⟩ = ⟨x, y⟩ → (x = 𝑋 y = 𝑌))
12112eximi 1476 . . . 4 (xy𝑋, 𝑌⟩ = ⟨x, y⟩ → xy(x = 𝑋 y = 𝑌))
136, 12syl 14 . . 3 (𝑋{⟨x, y⟩ ∣ φ}𝑌xy(x = 𝑋 y = 𝑌))
14 eeanv 1790 . . 3 (xy(x = 𝑋 y = 𝑌) ↔ (x x = 𝑋 y y = 𝑌))
1513, 14sylib 127 . 2 (𝑋{⟨x, y⟩ ∣ φ}𝑌 → (x x = 𝑋 y y = 𝑌))
16 isset 2538 . . 3 (𝑋 V ↔ x x = 𝑋)
17 isset 2538 . . 3 (𝑌 V ↔ y y = 𝑌)
1816, 17anbi12i 436 . 2 ((𝑋 V 𝑌 V) ↔ (x x = 𝑋 y y = 𝑌))
1915, 18sylibr 137 1 (𝑋{⟨x, y⟩ ∣ φ}𝑌 → (𝑋 V 𝑌 V))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1362   = wceq 1374   ∈ wcel 1376  Vcvv 2534  ⟨cop 3331   class class class wbr 3717  {copab 3770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-14 1388  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005  ax-sep 3828  ax-pow 3880  ax-pr 3897 This theorem depends on definitions:  df-bi 110  df-3an 878  df-tru 1232  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-un 2901  df-in 2903  df-ss 2910  df-pw 3314  df-sn 3334  df-pr 3335  df-op 3337  df-br 3718  df-opab 3772 This theorem is referenced by: (None)
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