Theorem brabvv 6134

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 4131	. . . . . 6 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2		elopab 4400	. . . . . 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 1670	. . . . . 6 ⊢ (∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
5	4	eximi 1653	. . . . 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 2824	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	7, 8	opth 4377	. . . . . . 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 2241	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
12	11	2eximi 1654	. . . 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 1992	. . 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 2828	. . 3 ⊢ (𝑋 ∈ V ↔ ∃𝑥 𝑥 = 𝑋)
17		isset 2828	. . 3 ⊢ (𝑌 ∈ V ↔ ∃𝑦 𝑦 = 𝑌)
18	16, 17	anbi12i 464	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ⟨cop 3712   class class class wbr 4130  {copab 4191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193

This theorem is used by: (None)