Theorem bj-elsn0 36543

Description: If the intersection of two classes is a set, then these classes are equal if and only if one is an element of the singleton formed on the other. Stronger form of elsng 4637 and elsn2g 4661 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.)

Assertion

Ref	Expression
bj-elsn0	⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem bj-elsn0

Step	Hyp	Ref	Expression
1		elsni 4640	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2		bj-inexeqex 36542	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
4		elsng 4637	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
5	4	biimprd 247	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵}))
6	2, 3, 5	3syl 18	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵}))
7	6	ex 412	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵})))
8	7	pm2.43d 53	. 2 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵}))
9	1, 8	impbid2 225	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∩ cin 3942  {csn 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-sn 4624

This theorem is referenced by: (None)