Theorem eltpg 4617

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 4582	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
2		elsng 4575	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷))
3	1, 2	orbi12d 915	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷)))
4		df-tp 4566	. . . 4 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
5	4	eleq2i 2904	. . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}))
6		elun 4125	. . 3 ⊢ (𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
7	5, 6	bitri 277	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
8		df-3or 1084	. 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷))
9	3, 7, 8	3bitr4g 316	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 843   ∨ w3o 1082   = wceq 1533   ∈ wcel 2110   ∪ cun 3934  {csn 4561  {cpr 4563  {ctp 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-un 3941  df-sn 4562  df-pr 4564  df-tp 4566

This theorem is referenced by:  eldiftp  4618  eltpi  4619  eltp  4620  tpid1g  4699  tpid2g  4701  tpid3g  4702  f1dom3fv3dif  7020  f1dom3el3dif  7021  lcmftp  15974  estrreslem2  17382  1cubr  25414  zabsle1  25866  nb3grprlem1  27156