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

Step	Hyp	Ref	Expression
1		elprg 3750	. . 3 ⊢ (A ∈ V → (A ∈ {B, C} ↔ (A = B ∨ A = C)))
2		elsncg 3755	. . 3 ⊢ (A ∈ V → (A ∈ {D} ↔ A = D))
3	1, 2	orbi12d 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})
5	4	eleq2i 2417	. . 3 ⊢ (A ∈ {B, C, D} ↔ A ∈ ({B, C} ∪ {D}))
6		elun 3220	. . 3 ⊢ (A ∈ ({B, C} ∪ {D}) ↔ (A ∈ {B, C} ∨ A ∈ {D}))
7	5, 6	bitri 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))
9	3, 7, 8	3bitr4g 279	1 ⊢ (A ∈ V → (A ∈ {B, C, D} ↔ (A = B ∨ A = C ∨ A = D)))

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-mp 5  ax-1 6  ax-2 7  ax-3 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