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Theorem eltpg 4691
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 4653 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
2 elsng 4645 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷))
31, 2orbi12d 918 . 2 (𝐴𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵𝐴 = 𝐶) ∨ 𝐴 = 𝐷)))
4 df-tp 4636 . . . 4 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
54eleq2i 2831 . . 3 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}))
6 elun 4163 . . 3 (𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
75, 6bitri 275 . 2 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
8 df-3or 1087 . 2 ((𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷) ↔ ((𝐴 = 𝐵𝐴 = 𝐶) ∨ 𝐴 = 𝐷))
93, 7, 83bitr4g 314 1 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 847  w3o 1085   = wceq 1537  wcel 2106  cun 3961  {csn 4631  {cpr 4633  {ctp 4635
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634  df-tp 4636
This theorem is referenced by:  eldiftp  4692  eltpi  4693  eltp  4694  el7g  4695  tpid1g  4774  tpid2g  4776  tpid3g  4777  f1dom3fv3dif  7288  f1dom3el3dif  7289  lcmftp  16670  estrreslem2  18194  1cubr  26900  zabsle1  27355  nb3grprlem1  29412
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