Theorem eldiftp 4693

Description: Membership in a set with three elements removed. Similar to eldifsn 4793 and eldifpr 4663. (Contributed by David A. Wheeler, 22-Jul-2017.)

Assertion

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

Proof of Theorem eldiftp

Step	Hyp	Ref	Expression
1		eldif 3957	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}))
2		eltpg 4692	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐸)))
3	2	notbid 317	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ ¬ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐸)))
4		ne3anior 3032	. . . 4 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸) ↔ ¬ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐸))
5	3, 4	bitr4di 288	. . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))
6	5	pm5.32i 573	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))
7	1, 6	bitri 274	1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 394   ∨ w3o 1083   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2936   ∖ cdif 3944  {ctp 4634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-v 3473  df-dif 3950  df-un 3952  df-sn 4631  df-pr 4633  df-tp 4635

This theorem is referenced by: (None)