Theorem eltpg 4686

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 4648	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
2		elsng 4640	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷))
3	1, 2	orbi12d 919	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷)))
4		df-tp 4631	. . . 4 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
5	4	eleq2i 2833	. . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}))
6		elun 4153	. . 3 ⊢ (𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
7	5, 6	bitri 275	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
8		df-3or 1088	. 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷))
9	3, 7, 8	3bitr4g 314	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 848   ∨ w3o 1086   = wceq 1540   ∈ wcel 2108   ∪ cun 3949  {csn 4626  {cpr 4628  {ctp 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629  df-tp 4631

This theorem is referenced by:  eldiftp  4687  eltpi  4688  eltp  4689  el7g  4690  tpid1g  4769  tpid2g  4771  tpid3g  4772  f1dom3fv3dif  7288  f1dom3el3dif  7289  lcmftp  16673  estrreslem2  18183  1cubr  26885  zabsle1  27340  nb3grprlem1  29397