Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eldiftp Structured version   Visualization version   GIF version

Theorem eldiftp 4206
 Description: Membership in a set with three elements removed. Similar to eldifsn 4294 and eldifpr 4182. (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 3570 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}))
2 eltpg 4205 . . . . 5 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
32notbid 308 . . . 4 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸)))
4 ne3anior 2883 . . . 4 ((𝐴𝐶𝐴𝐷𝐴𝐸) ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐸))
53, 4syl6bbr 278 . . 3 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴𝐶𝐴𝐷𝐴𝐸)))
65pm5.32i 668 . 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 384   ∨ w3o 1035   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790   ∖ cdif 3557  {ctp 4159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3192  df-dif 3563  df-un 3565  df-sn 4156  df-pr 4158  df-tp 4160 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator