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Theorem eldifpr 4661
Description: Membership in a set with two elements removed. Similar to eldifsn 4791 and eldiftp 4691. (Contributed by Mario Carneiro, 18-Jul-2017.)
Assertion
Ref Expression
eldifpr (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Proof of Theorem eldifpr
StepHypRef Expression
1 elprg 4650 . . . . 5 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷)))
21notbid 318 . . . 4 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷)))
3 neanior 3036 . . . 4 ((𝐴𝐶𝐴𝐷) ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷))
42, 3bitr4di 289 . . 3 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴𝐶𝐴𝐷)))
54pm5.32i 576 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷)))
6 eldif 3959 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}))
7 3anass 1096 . 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  cdif 3946  {cpr 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-un 3954  df-sn 4630  df-pr 4632
This theorem is referenced by:  rexdifpr  4662  logbcl  26272  logbid1  26273  logb1  26274  elogb  26275  logbchbase  26276  relogbval  26277  relogbcl  26278  relogbreexp  26280  relogbmul  26282  relogbexp  26285  nnlogbexp  26286  relogbcxp  26290  cxplogb  26291  relogbcxpb  26292  logbmpt  26293  logbfval  26295  logbgt0b  26298  2logb9irrALT  26303  sqrt2cxp2logb9e3  26304  neldifpr1  31770  neldifpr2  31771  eluz2cnn0n1  47192  rege1logbrege0  47244  relogbmulbexp  47247  relogbdivb  47248  nnpw2blen  47266
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