Theorem bj-brrelex12ALT 37435

Description: Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5673. (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 5681	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
2		jcn 162	. . . 4 ⊢ (𝐴𝑅𝐵 → (¬ ∅ ∈ 𝑅 → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅)))
3	2	impcom 409	. . 3 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅))
4		opprc 4830	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
5		df-br 5076	. . . . . 6 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
6	5	biimpi 218	. . . . 5 ⊢ (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
7		eleq1 2829	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ∅ ∈ 𝑅))
8	6, 7	imbitrid 246	. . . 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 587	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433  ∅c0 4264  ⟨cop 4564   class class class wbr 5075  Rel wrel 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628

This theorem is referenced by: (None)