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Theorem eltpg 3769
 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 (A V → (A {B, C, D} ↔ (A = B A = C A = D)))

Proof of Theorem eltpg
StepHypRef Expression
1 elprg 3750 . . 3 (A V → (A {B, C} ↔ (A = B A = C)))
2 elsncg 3755 . . 3 (A V → (A {D} ↔ A = D))
31, 2orbi12d 690 . 2 (A V → ((A {B, C} A {D}) ↔ ((A = B A = C) A = D)))
4 df-tp 3743 . . . 4 {B, C, D} = ({B, C} ∪ {D})
54eleq2i 2417 . . 3 (A {B, C, D} ↔ A ({B, C} ∪ {D}))
6 elun 3220 . . 3 (A ({B, C} ∪ {D}) ↔ (A {B, C} A {D}))
75, 6bitri 240 . 2 (A {B, C, D} ↔ (A {B, C} A {D}))
8 df-3or 935 . 2 ((A = B A = C A = D) ↔ ((A = B A = C) A = D))
93, 7, 83bitr4g 279 1 (A V → (A {B, C, D} ↔ (A = B A = C A = D)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∨ w3o 933   = wceq 1642   ∈ wcel 1710   ∪ cun 3207  {csn 3737  {cpr 3738  {ctp 3739 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743 This theorem is referenced by:  eltpi  3770  eltp  3771
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