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Theorem bj-elsn0 36339
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 4641 and elsn2g 4665 (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 4644 . 2 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2 bj-inexeqex 36338 . . . . 5 (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 simpl 481 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
4 elsng 4641 . . . . . 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 411 . . 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 394   = wceq 1539  wcel 2104  Vcvv 3472  cin 3946  {csn 4627
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-in 3954  df-ss 3964  df-sn 4628
This theorem is referenced by: (None)
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