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Theorem eqoreldif 4582
 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 486 . . . . 5 ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐶)
2 elsni 4542 . . . . . . 7 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
32con3i 157 . . . . . 6 𝐴 = 𝐵 → ¬ 𝐴 ∈ {𝐵})
43adantl 485 . . . . 5 ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 ∈ {𝐵})
51, 4eldifd 3871 . . . 4 ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ (𝐶 ∖ {𝐵}))
65ex 416 . . 3 (𝐴𝐶 → (¬ 𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})))
76orrd 860 . 2 (𝐴𝐶 → (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})))
8 eleq1a 2847 . . 3 (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶))
9 eldifi 4034 . . . 4 (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴𝐶)
109a1i 11 . . 3 (𝐵𝐶 → (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴𝐶))
118, 10jaod 856 . 2 (𝐵𝐶 → ((𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})) → 𝐴𝐶))
127, 11impbid2 229 1 (𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ∖ cdif 3857  {csn 4525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-dif 3863  df-sn 4526 This theorem is referenced by:  lcmfunsnlem2  16050
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