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Theorem eqoreldif 4216
Description: An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.) (Proof shortened by JJ, 23-Jul-2021.)
Assertion
Ref Expression
eqoreldif (𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))

Proof of Theorem eqoreldif
StepHypRef Expression
1 simpl 473 . . . . 5 ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐶)
2 elsni 4185 . . . . . . 7 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
32con3i 150 . . . . . 6 𝐴 = 𝐵 → ¬ 𝐴 ∈ {𝐵})
43adantl 482 . . . . 5 ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 ∈ {𝐵})
51, 4eldifd 3578 . . . 4 ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ (𝐶 ∖ {𝐵}))
65ex 450 . . 3 (𝐴𝐶 → (¬ 𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})))
76orrd 393 . 2 (𝐴𝐶 → (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})))
8 eleq1a 2694 . . 3 (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶))
9 eldifi 3724 . . . 4 (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴𝐶)
109a1i 11 . . 3 (𝐵𝐶 → (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴𝐶))
118, 10jaod 395 . 2 (𝐵𝐶 → ((𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})) → 𝐴𝐶))
127, 11impbid2 216 1 (𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1481  wcel 1988  cdif 3564  {csn 4168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570  df-sn 4169
This theorem is referenced by:  lcmfunsnlem2  15334
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