Theorem eldifpr 4616

Description: Membership in a set with two elements removed. Similar to eldifsn 4743 and eldiftp 4645. (Contributed by Mario Carneiro, 18-Jul-2017.)

Assertion

Ref	Expression
eldifpr	⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))

Proof of Theorem eldifpr

Step	Hyp	Ref	Expression
1		elprg 4604	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
2	1	notbid 318	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ ¬ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
3		neanior 3026	. . . 4 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
4	2, 3	bitr4di 289	. . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)))
5	4	pm5.32i 574	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)))
6		eldif 3912	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}))
7		3anass 1095	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)))
8	5, 6, 7	3bitr4i 303	1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∖ cdif 3899  {cpr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3443  df-dif 3905  df-un 3907  df-sn 4582  df-pr 4584

This theorem is referenced by:  rexdifpr  4617  logbcl  26737  logbid1  26738  logb1  26739  elogb  26740  logbchbase  26741  relogbval  26742  relogbcl  26743  relogbreexp  26745  relogbmul  26747  relogbexp  26750  nnlogbexp  26751  relogbcxp  26755  cxplogb  26756  relogbcxpb  26757  logbmpt  26758  logbfval  26760  logbgt0b  26763  2logb9irrALT  26768  sqrt2cxp2logb9e3  26769  neldifpr1  32611  neldifpr2  32612  eluz2cnn0n1  48824  rege1logbrege0  48871  relogbmulbexp  48874  relogbdivb  48875  nnpw2blen  48893