Theorem brabv 5568

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

Step	Hyp	Ref	Expression
1		df-br 5148	. 2 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2		opprc 4895	. . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨𝑋, 𝑌⟩ = ∅)
3		0nelopab 5566	. . . . 5 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
4		eleq1 2821	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = ∅ → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
5	3, 4	mtbiri 326	. . . 4 ⊢ (⟨𝑋, 𝑌⟩ = ∅ → ¬ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6	2, 5	syl 17	. . 3 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ¬ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
7	6	con4i 114	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V))
8	1, 7	sylbi 216	1 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∅c0 4321  ⟨cop 4633   class class class wbr 5147  {copab 5209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210

This theorem is referenced by:  brfvopab  7462  bropopvvv  8072  bropfvvvvlem  8073  isfunc  17810  eqgval  19051  bj-imdirval3  36053  upwlkbprop  46502