Theorem brabv 6699

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 4654	. 2 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2		opprc 4424	. . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨𝑋, 𝑌⟩ = ∅)
3		0neqopab 6698	. . . . 5 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
4		eleq1 2689	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = ∅ → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
5	3, 4	mtbiri 317	. . . 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 113	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V))
8	1, 7	sylbi 207	1 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  ⟨cop 4183   class class class wbr 4653  {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713

This theorem is referenced by:  brfvopab  6700  bropopvvv  7255  bropfvvvvlem  7256  isfunc  16524  eqgval  17643  rgrprop  26456  rusgrprop  26458  upwlkbprop  41719