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Theorem eldiftp 4691
Description: Membership in a set with three elements removed. Similar to eldifsn 4791 and eldifpr 4661. (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 3957 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}))
2 eltpg 4690 . . . . 5 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
32notbid 318 . . . 4 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
4 ne3anior 3033 . . . 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 1084  w3a 1085   = wceq 1534  wcel 2099  wne 2937  cdif 3944  {ctp 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3473  df-dif 3950  df-un 3952  df-sn 4630  df-pr 4632  df-tp 4634
This theorem is referenced by: (None)
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