Theorem bj-elsn0 37894

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 4601 and elsn2g 4628 (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 4604	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2		bj-inexeqex 37893	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		simpl 488	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
4		elsng 4601	. . . . . 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 418	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∩ cin 3901  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-in 3909  df-ss 3919  df-sn 4588

This theorem is used by: (None)