Theorem brabvv 5923

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 4006	. . . . . 6 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2		elopab 4260	. . . . . 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 1617	. . . . . 6 ⊢ (∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
5	4	eximi 1600	. . . . 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 2742	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 2742	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	7, 8	opth 4239	. . . . . . 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 2180	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
12	11	2eximi 1601	. . . 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 1932	. . 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 2745	. . 3 ⊢ (𝑋 ∈ V ↔ ∃𝑥 𝑥 = 𝑋)
17		isset 2745	. . 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 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2739  ⟨cop 3597   class class class wbr 4005  {copab 4065

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067

This theorem is referenced by: (None)