Theorem elprneb 41296

Description: An element of a proper unordered pair is the first element iff it is not the second element. (Contributed by AV, 18-Jun-2020.)

Assertion

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

Proof of Theorem elprneb

Step	Hyp	Ref	Expression
1		elpri 4197	. . 3 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
2		neeq1 2856	. . . . . 6 |- ( B = A -> ( B =/= C <-> A =/= C ) )
3	2	eqcoms 2630	. . . . 5 |- ( A = B -> ( B =/= C <-> A =/= C ) )
4		pm5.1 902	. . . . . 6 |- ( ( A = B /\ A =/= C ) -> ( A = B <-> A =/= C ) )
5	4	ex 450	. . . . 5 |- ( A = B -> ( A =/= C -> ( A = B <-> A =/= C ) ) )
6	3, 5	sylbid 230	. . . 4 |- ( A = B -> ( B =/= C -> ( A = B <-> A =/= C ) ) )
7		neeq2 2857	. . . . 5 |- ( A = C -> ( B =/= A <-> B =/= C ) )
8		nesym 2850	. . . . . . . 8 |- ( B =/= A <-> -. A = B )
9		pm5.1 902	. . . . . . . 8 |- ( ( A = C /\ -. A = B ) -> ( A = C <-> -. A = B ) )
10	8, 9	sylan2b 492	. . . . . . 7 |- ( ( A = C /\ B =/= A ) -> ( A = C <-> -. A = B ) )
11	10	necon2abid 2836	. . . . . 6 |- ( ( A = C /\ B =/= A ) -> ( A = B <-> A =/= C ) )
12	11	ex 450	. . . . 5 |- ( A = C -> ( B =/= A -> ( A = B <-> A =/= C ) ) )
13	7, 12	sylbird 250	. . . 4 |- ( A = C -> ( B =/= C -> ( A = B <-> A =/= C ) ) )
14	6, 13	jaoi 394	. . 3 |- ( ( A = B \/ A = C ) -> ( B =/= C -> ( A = B <-> A =/= C ) ) )
15	1, 14	syl 17	. 2 |- ( A e. { B , C } -> ( B =/= C -> ( A = B <-> A =/= C ) ) )
16	15	imp 445	1 |- ( ( A e. { B , C } /\ B =/= C ) -> ( A = B <-> A =/= C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ 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:  dfodd5  41572