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Theorem eldiftp 4366
 Description: Membership in a set with three elements removed. Similar to eldifsn 4454 and eldifpr 4344. (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 3733 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}))
2 eltpg 4365 . . . . 5 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
32notbid 307 . . . 4 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
4 ne3anior 3036 . . . 4 ((𝐴𝐶𝐴𝐷𝐴𝐸) ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸))
53, 4syl6bbr 278 . . 3 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴𝐶𝐴𝐷𝐴𝐸)))
65pm5.32i 564 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))
71, 6bitri 264 1 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 382   ∨ w3o 1070   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943   ∖ cdif 3720  {ctp 4321 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-sn 4318  df-pr 4320  df-tp 4322 This theorem is referenced by: (None)
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