Theorem brabvv 5551

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 3765	. . . . . 6 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2		elopab 3995	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3	1, 2	bitri 173	. . . . 5 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		exsimpl 1508	. . . . . 6 ⊢ (∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
5	4	eximi 1491	. . . . 5 ⊢ (∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
6	3, 5	sylbi 114	. . . 4 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → ∃𝑥∃𝑦⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
7		vex 2560	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 2560	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	7, 8	opth 3974	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ ↔ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
10	9	biimpi 113	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ → (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
11	10	eqcoms 2043	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
12	11	2eximi 1492	. . . 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 1807	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑋 ∧ 𝑦 = 𝑌) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
15	13, 14	sylib 127	. 2 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
16		isset 2561	. . 3 ⊢ (𝑋 ∈ V ↔ ∃𝑥 𝑥 = 𝑋)
17		isset 2561	. . 3 ⊢ (𝑌 ∈ V ↔ ∃𝑦 𝑦 = 𝑌)
18	16, 17	anbi12i 433	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
19	15, 18	sylibr 137	1 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378   class class class wbr 3764  {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819

This theorem is referenced by: (None)