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Theorem eltpg 4364
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 4336 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
2 elsng 4330 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷))
31, 2orbi12d 904 . 2 (𝐴𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵𝐴 = 𝐶) ∨ 𝐴 = 𝐷)))
4 df-tp 4321 . . . 4 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
54eleq2i 2842 . . 3 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}))
6 elun 3904 . . 3 (𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
75, 6bitri 264 . 2 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
8 df-3or 1072 . 2 ((𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷) ↔ ((𝐴 = 𝐵𝐴 = 𝐶) ∨ 𝐴 = 𝐷))
93, 7, 83bitr4g 303 1 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 836  w3o 1070   = wceq 1631  wcel 2145  cun 3721  {csn 4316  {cpr 4318  {ctp 4320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728  df-sn 4317  df-pr 4319  df-tp 4321
This theorem is referenced by:  eldiftp  4365  eltpi  4366  eltp  4367  tpid3g  4441  f1dom3fv3dif  6668  f1dom3el3dif  6669  lcmftp  15557  estrreslem2  16985  1cubr  24790  zabsle1  25242  nb3grprlem1  26505  tpid2g  39837  tpid1g  39843
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