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Theorem eldifpr 4623
Description: Membership in a set with two elements removed. Similar to eldifsn 4752 and eldiftp 4652. (Contributed by Mario Carneiro, 18-Jul-2017.)
Assertion
Ref Expression
eldifpr (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Proof of Theorem eldifpr
StepHypRef Expression
1 elprg 4612 . . . . 5 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷)))
21notbid 318 . . . 4 (𝐴𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ ¬ (𝐴 = 𝐶𝐴 = 𝐷)))
3 neanior 3038 . . . 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 3925 . 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 2944  cdif 3912  {cpr 4593
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 2708
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3450  df-dif 3918  df-un 3920  df-sn 4592  df-pr 4594
This theorem is referenced by:  rexdifpr  4624  logbcl  26133  logbid1  26134  logb1  26135  elogb  26136  logbchbase  26137  relogbval  26138  relogbcl  26139  relogbreexp  26141  relogbmul  26143  relogbexp  26146  nnlogbexp  26147  relogbcxp  26151  cxplogb  26152  relogbcxpb  26153  logbmpt  26154  logbfval  26156  logbgt0b  26159  2logb9irrALT  26164  sqrt2cxp2logb9e3  26165  neldifpr1  31502  neldifpr2  31503  eluz2cnn0n1  46666  rege1logbrege0  46718  relogbmulbexp  46721  relogbdivb  46722  nnpw2blen  46740
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