Theorem eltpi 3756

Description: A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
eltpi	|- ( A e. { B , C , D } -> ( A = B \/ A = C \/ A = D ) )

Proof of Theorem eltpi

Step	Hyp	Ref	Expression
1		eltpg 3754	. 2 |- ( A e. { B , C , D } -> ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) )
2	1	ibi 176	1 |- ( A e. { B , C , D } -> ( A = B \/ A = C \/ A = D ) )

This proof depends on syntax axioms:   -> wi 4   \/ w3o 1008   = wceq 1402   e. wcel 2209   {ctp 3711

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3or 1010  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-tp 3717

This theorem is used by:  prm23lt5  13042  perfectlem2  16114  zabsle1  16118