Theorem eltpg 3733

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 3708	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
2		elsng 3703	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷))
3	1, 2	orbi12d 801	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷)))
4		df-tp 3696	. . . 4 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
5	4	eleq2i 2299	. . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}))
6		elun 3359	. . 3 ⊢ (𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
7	5, 6	bitri 184	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
8		df-3or 1006	. 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷))
9	3, 7, 8	3bitr4g 223	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 716   ∨ w3o 1004   = wceq 1398   ∈ wcel 2203   ∪ cun 3208  {csn 3688  {cpr 3689  {ctp 3690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-tp 3696

This theorem is referenced by:  eldiftp  3734  eltpi  3735  eltp  3736  tpid1g  3803  tpid2g  3805  zabsle1  15859  gausslemma2dlem0i  15917  2lgsoddprm  15973