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Theorem brabvv 5493
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 3756 . . . . . 6 (𝑋{⟨x, y⟩ ∣ φ}𝑌 ↔ ⟨𝑋, 𝑌 {⟨x, y⟩ ∣ φ})
2 elopab 3986 . . . . . 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 1505 . . . . . 6 (y(⟨𝑋, 𝑌⟩ = ⟨x, y φ) → y𝑋, 𝑌⟩ = ⟨x, y⟩)
54eximi 1488 . . . . 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 2554 . . . . . . . 8 x V
8 vex 2554 . . . . . . . 8 y V
97, 8opth 3965 . . . . . . 7 (⟨x, y⟩ = ⟨𝑋, 𝑌⟩ ↔ (x = 𝑋 y = 𝑌))
109biimpi 113 . . . . . 6 (⟨x, y⟩ = ⟨𝑋, 𝑌⟩ → (x = 𝑋 y = 𝑌))
1110eqcoms 2040 . . . . 5 (⟨𝑋, 𝑌⟩ = ⟨x, y⟩ → (x = 𝑋 y = 𝑌))
12112eximi 1489 . . . 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 1804 . . 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 2555 . . 3 (𝑋 V ↔ x x = 𝑋)
17 isset 2555 . . 3 (𝑌 V ↔ y y = 𝑌)
1816, 17anbi12i 433 . 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   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370   class class class wbr 3755  {copab 3808
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810
This theorem is referenced by: (None)
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