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Theorem brabv 6741
Description: If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
Assertion
Ref Expression
brabv (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Proof of Theorem brabv
StepHypRef Expression
1 df-br 4686 . 2 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2 opprc 4456 . . . 4 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨𝑋, 𝑌⟩ = ∅)
3 0neqopab 6740 . . . . 5 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
4 eleq1 2718 . . . . 5 (⟨𝑋, 𝑌⟩ = ∅ → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
53, 4mtbiri 316 . . . 4 (⟨𝑋, 𝑌⟩ = ∅ → ¬ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
62, 5syl 17 . . 3 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ¬ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
76con4i 113 . 2 (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V))
81, 7sylbi 207 1 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  cop 4216   class class class wbr 4685  {copab 4745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746
This theorem is referenced by:  brfvopab  6742  bropopvvv  7300  bropfvvvvlem  7301  isfunc  16571  eqgval  17690  rgrprop  26512  rusgrprop  26514  upwlkbprop  42044
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