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Theorem bj-elsn0 36036
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 4643 and elsn2g 4667 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.)
Assertion
Ref Expression
bj-elsn0 ((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem bj-elsn0
StepHypRef Expression
1 elsni 4646 . 2 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2 bj-inexeqex 36035 . . . . 5 (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 simpl 484 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
4 elsng 4643 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
54biimprd 247 . . . . 5 (𝐴 ∈ V → (𝐴 = 𝐵𝐴 ∈ {𝐵}))
62, 3, 53syl 18 . . . 4 (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → (𝐴 = 𝐵𝐴 ∈ {𝐵}))
76ex 414 . . 3 ((𝐴𝐵) ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵𝐴 ∈ {𝐵})))
87pm2.43d 53 . 2 ((𝐴𝐵) ∈ 𝑉 → (𝐴 = 𝐵𝐴 ∈ {𝐵}))
91, 8impbid2 225 1 ((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cin 3948  {csn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-sn 4630
This theorem is referenced by: (None)
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