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Theorem eltpg 3636
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 3611 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
2 elsng 3606 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷))
31, 2orbi12d 793 . 2 (𝐴𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵𝐴 = 𝐶) ∨ 𝐴 = 𝐷)))
4 df-tp 3599 . . . 4 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
54eleq2i 2244 . . 3 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}))
6 elun 3276 . . 3 (𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
75, 6bitri 184 . 2 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}))
8 df-3or 979 . 2 ((𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷) ↔ ((𝐴 = 𝐵𝐴 = 𝐶) ∨ 𝐴 = 𝐷))
93, 7, 83bitr4g 223 1 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 708  w3o 977   = wceq 1353  wcel 2148  cun 3127  {csn 3591  {cpr 3592  {ctp 3593
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3or 979  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-tp 3599
This theorem is referenced by:  eldiftp  3637  eltpi  3638  eltp  3639  tpid1g  3703  tpid2g  3705  zabsle1  14060
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