Theorem bj-elsn0 37743

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 4602 and elsn2g 4629 (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 4605	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2		bj-inexeqex 37742	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		simpl 487	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
4		elsng 4602	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
5	4	biimprd 251	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵}))
6	2, 3, 5	3syl 19	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵}))
7	6	ex 417	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵})))
8	7	pm2.43d 54	. 2 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵}))
9	1, 8	impbid2 229	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  Vcvv 3453   ∩ cin 3903  {csn 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-in 3911  df-ss 3921  df-sn 4589

This theorem is referenced by: (None)