Theorem bj-elsn0 34471

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 4574 and elsn2g 4596 (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 4577	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2		bj-inexeqex 34470	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		simpl 485	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
4		elsng 4574	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
5	4	biimprd 250	. . . . 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 415	. . 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 228	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3491   ∩ cin 3928  {csn 4560

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3493  df-in 3936  df-ss 3945  df-sn 4561

This theorem is referenced by: (None)