Theorem brabvv 6077

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

Step	Hyp	Ref	Expression
1		df-br 4094	. . . . . 6 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2		elopab 4358	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3	1, 2	bitri 184	. . . . 5 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		exsimpl 1666	. . . . . 6 ⊢ (∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
5	4	eximi 1649	. . . . 5 ⊢ (∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
6	3, 5	sylbi 121	. . . 4 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → ∃𝑥∃𝑦⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
7		vex 2806	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 2806	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	7, 8	opth 4335	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ ↔ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
10	9	biimpi 120	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ → (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
11	10	eqcoms 2234	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
12	11	2eximi 1650	. . . 4 ⊢ (∃𝑥∃𝑦⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦(𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
13	6, 12	syl 14	. . 3 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → ∃𝑥∃𝑦(𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
14		eeanv 1985	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑋 ∧ 𝑦 = 𝑌) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
15	13, 14	sylib 122	. 2 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
16		isset 2810	. . 3 ⊢ (𝑋 ∈ V ↔ ∃𝑥 𝑥 = 𝑋)
17		isset 2810	. . 3 ⊢ (𝑌 ∈ V ↔ ∃𝑦 𝑦 = 𝑌)
18	16, 17	anbi12i 460	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
19	15, 18	sylibr 134	1 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803  ⟨cop 3676   class class class wbr 4093  {copab 4154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156

This theorem is referenced by: (None)