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Theorem eldifpr 4656
Description: Membership in a set with two elements removed. Similar to eldifsn 4786 and eldiftp 4686. (Contributed by Mario Carneiro, 18-Jul-2017.)
Assertion
Ref Expression
eldifpr (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Proof of Theorem eldifpr
StepHypRef Expression
1 elprg 4645 . . . . 5 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷)))
21notbid 318 . . . 4 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷)))
3 neanior 3031 . . . 4 ((𝐴𝐶𝐴𝐷) ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷))
42, 3bitr4di 289 . . 3 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴𝐶𝐴𝐷)))
54pm5.32i 574 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷)))
6 eldif 3955 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}))
7 3anass 1093 . 2 ((𝐴𝐵𝐴𝐶𝐴𝐷) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷)))
85, 6, 73bitr4i 303 1 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  wne 2936  cdif 3942  {cpr 4626
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-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-v 3472  df-dif 3948  df-un 3950  df-sn 4625  df-pr 4627
This theorem is referenced by:  rexdifpr  4657  logbcl  26692  logbid1  26693  logb1  26694  elogb  26695  logbchbase  26696  relogbval  26697  relogbcl  26698  relogbreexp  26700  relogbmul  26702  relogbexp  26705  nnlogbexp  26706  relogbcxp  26710  cxplogb  26711  relogbcxpb  26712  logbmpt  26713  logbfval  26715  logbgt0b  26718  2logb9irrALT  26723  sqrt2cxp2logb9e3  26724  neldifpr1  32322  neldifpr2  32323  eluz2cnn0n1  47573  rege1logbrege0  47625  relogbmulbexp  47628  relogbdivb  47629  nnpw2blen  47647
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