Theorem bj-brrelex12ALT 37080

Description: Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5666. (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 5674	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
2		jcn 162	. . . 4 ⊢ (𝐴𝑅𝐵 → (¬ ∅ ∈ 𝑅 → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅)))
3	2	impcom 407	. . 3 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅))
4		opprc 4846	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
5		df-br 5090	. . . . . 6 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
6	5	biimpi 216	. . . . 5 ⊢ (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
7		eleq1 2817	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ∅ ∈ 𝑅))
8	6, 7	imbitrid 244	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (𝐴𝑅𝐵 → ∅ ∈ 𝑅))
9	4, 8	syl 17	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 → ∅ ∈ 𝑅))
10	3, 9	nsyl2 141	. 2 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
11	1, 10	sylan 580	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  Vcvv 3434  ∅c0 4281  ⟨cop 4580   class class class wbr 5089  Rel wrel 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621

This theorem is referenced by: (None)