Theorem bj-brrelex12ALT 34383

Description: Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5597. (Contributed by BJ, 14-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-brrelex12ALT	⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem bj-brrelex12ALT

Step	Hyp	Ref	Expression
1		0nelrel0 5605	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
2		jcn 164	. . . 4 ⊢ (𝐴𝑅𝐵 → (¬ ∅ ∈ 𝑅 → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅)))
3	2	impcom 410	. . 3 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅))
4		opprc 4819	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
5		df-br 5060	. . . . . 6 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
6	5	biimpi 218	. . . . 5 ⊢ (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
7		eleq1 2899	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ∅ ∈ 𝑅))
8	6, 7	syl5ib 246	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (𝐴𝑅𝐵 → ∅ ∈ 𝑅))
9	4, 8	syl 17	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 → ∅ ∈ 𝑅))
10	3, 9	nsyl2 143	. 2 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
11	1, 10	sylan 582	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3491  ∅c0 4284  ⟨cop 4566   class class class wbr 5059  Rel wrel 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122  df-xp 5554  df-rel 5555

This theorem is referenced by: (None)