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Theorem eldiftp 4685
Description: Membership in a set with three elements removed. Similar to eldifsn 4785 and eldifpr 4655. (Contributed by David A. Wheeler, 22-Jul-2017.)
Assertion
Ref Expression
eldiftp (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))

Proof of Theorem eldiftp
StepHypRef Expression
1 eldif 3953 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}))
2 eltpg 4684 . . . . 5 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
32notbid 318 . . . 4 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
4 ne3anior 3030 . . . 4 ((𝐴𝐶𝐴𝐷𝐴𝐸) ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸))
53, 4bitr4di 289 . . 3 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴𝐶𝐴𝐷𝐴𝐸)))
65pm5.32i 574 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))
71, 6bitri 275 1 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  w3o 1083  w3a 1084   = wceq 1533  wcel 2098  wne 2934  cdif 3940  {ctp 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-v 3470  df-dif 3946  df-un 3948  df-sn 4624  df-pr 4626  df-tp 4628
This theorem is referenced by: (None)
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