Theorem bj-brrelex12ALT 37085

Description: Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5706. (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 5714	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
2		jcn 162	. . . 4 ⊢ (𝐴𝑅𝐵 → (¬ ∅ ∈ 𝑅 → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅)))
3	2	impcom 407	. . 3 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅))
4		opprc 4872	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
5		df-br 5120	. . . . . 6 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
6	5	biimpi 216	. . . . 5 ⊢ (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
7		eleq1 2822	. . . . 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 1540   ∈ wcel 2108  Vcvv 3459  ∅c0 4308  ⟨cop 4607   class class class wbr 5119  Rel wrel 5659

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661

This theorem is referenced by: (None)