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Theorem eltpg 3483
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 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))

Proof of Theorem eltpg
StepHypRef Expression
1 elprg 3461 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
2 elsng 3456 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷))
31, 2orbi12d 742 . 2 (𝐴𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵𝐴 = 𝐶) ∨ 𝐴 = 𝐷)))
4 df-tp 3449 . . . 4 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
54eleq2i 2154 . . 3 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}))
6 elun 3139 . . 3 (𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
75, 6bitri 182 . 2 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
8 df-3or 925 . 2 ((𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷) ↔ ((𝐴 = 𝐵𝐴 = 𝐶) ∨ 𝐴 = 𝐷))
93, 7, 83bitr4g 221 1 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 664  w3o 923   = wceq 1289  wcel 1438  cun 2995  {csn 3441  {cpr 3442  {ctp 3443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3or 925  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-tp 3449
This theorem is referenced by:  eltpi  3484  eltp  3485
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