Theorem eldiftp 4710

Description: Membership in a set with three elements removed. Similar to eldifsn 4811 and eldifpr 4680. (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 3986	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}))
2		eltpg 4709	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐸)))
3	2	notbid 318	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ ¬ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐸)))
4		ne3anior 3042	. . . 4 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸) ↔ ¬ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐸))
5	3, 4	bitr4di 289	. . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸} ↔ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))
6	5	pm5.32i 574	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))
7	1, 6	bitri 275	1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ w3o 1086   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∖ cdif 3973  {ctp 4652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-un 3981  df-sn 4649  df-pr 4651  df-tp 4653

This theorem is referenced by: (None)