Theorem elprn2 39866

Description: A member of an unordered pair that is not the "second", must be the "first". (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
elprn2	|- ( ( A e. { B , C } /\ A =/= C ) -> A = B )

Proof of Theorem elprn2

Step	Hyp	Ref	Expression
1		neneq 2800	. . 3 |- ( A =/= C -> -. A = C )
2	1	adantl 482	. 2 |- ( ( A e. { B , C } /\ A =/= C ) -> -. A = C )
3		elpri 4197	. . . 4 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
4	3	adantr 481	. . 3 |- ( ( A e. { B , C } /\ A =/= C ) -> ( A = B \/ A = C ) )
5		orcom 402	. . . 4 |- ( ( A = B \/ A = C ) <-> ( A = C \/ A = B ) )
6		df-or 385	. . . 4 |- ( ( A = C \/ A = B ) <-> ( -. A = C -> A = B ) )
7	5, 6	bitri 264	. . 3 |- ( ( A = B \/ A = C ) <-> ( -. A = C -> A = B ) )
8	4, 7	sylib 208	. 2 |- ( ( A e. { B , C } /\ A =/= C ) -> ( -. A = C -> A = B ) )
9	2, 8	mpd 15	1 |- ( ( A e. { B , C } /\ A =/= C ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)