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Theorem eldiftp 3617
Description: Membership in a set with three elements removed. Similar to eldifsn 3698 and eldifpr 3598. (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 3121 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}))
2 eltpg 3616 . . . . 5 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
32notbid 657 . . . 4 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
4 ne3anior 2422 . . . 4 ((𝐴𝐶𝐴𝐷𝐴𝐸) ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸))
53, 4bitr4di 197 . . 3 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴𝐶𝐴𝐷𝐴𝐸)))
65pm5.32i 450 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))
71, 6bitri 183 1 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  w3o 966  w3a 967   = wceq 1342  wcel 2135  wne 2334  cdif 3109  {ctp 3573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-v 2724  df-dif 3114  df-un 3116  df-sn 3577  df-pr 3578  df-tp 3579
This theorem is referenced by: (None)
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