Theorem eltpg 4690

Description: Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
eltpg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))

Proof of Theorem eltpg

Step	Hyp	Ref	Expression
1		elprg 4651	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
2		elsng 4643	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷))
3	1, 2	orbi12d 916	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷)))
4		df-tp 4634	. . . 4 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
5	4	eleq2i 2817	. . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}))
6		elun 4146	. . 3 ⊢ (𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
7	5, 6	bitri 274	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
8		df-3or 1085	. 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷))
9	3, 7, 8	3bitr4g 313	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 845   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098   ∪ cun 3943  {csn 4629  {cpr 4631  {ctp 4633

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 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-un 3950  df-sn 4630  df-pr 4632  df-tp 4634

This theorem is referenced by:  eldiftp  4691  eltpi  4692  eltp  4693  tpid1g  4774  tpid2g  4776  tpid3g  4777  f1dom3fv3dif  7276  f1dom3el3dif  7277  lcmftp  16606  estrreslem2  18128  1cubr  26804  zabsle1  27259  nb3grprlem1  29249