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Theorem eltpg 4198
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 4167 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
2 elsng 4162 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷))
31, 2orbi12d 745 . 2 (𝐴𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵𝐴 = 𝐶) ∨ 𝐴 = 𝐷)))
4 df-tp 4153 . . . 4 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
54eleq2i 2690 . . 3 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}))
6 elun 3731 . . 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 1037 . 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 383  w3o 1035   = wceq 1480  wcel 1987  cun 3553  {csn 4148  {cpr 4150  {ctp 4152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560  df-sn 4149  df-pr 4151  df-tp 4153
This theorem is referenced by:  eldiftp  4199  eltpi  4200  eltp  4201  tpid3g  4275  f1dom3fv3dif  6479  f1dom3el3dif  6480  lcmftp  15273  estrreslem2  16699  1cubr  24469  zabsle1  24921  nb3grprlem1  26169  tpid2g  38802  tpid1g  38809
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