Theorem nelpr1 39262

Description: If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
nelpr1.a	|- ( ph -> A e. V )
nelpr1.n	|- ( ph -> -. A e. { B , C } )

Assertion

Ref	Expression
nelpr1	|- ( ph -> A =/= B )

Proof of Theorem nelpr1

Step	Hyp	Ref	Expression
1		nelpr1.n	. . 3 |- ( ph -> -. A e. { B , C } )
2		animorrl 508	. . . 4 |- ( ( ph /\ A = B ) -> ( A = B \/ A = C ) )
3		nelpr1.a	. . . . . 6 |- ( ph -> A e. V )
4		elprg 4196	. . . . . 6 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
5	3, 4	syl 17	. . . . 5 |- ( ph -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
6	5	adantr 481	. . . 4 |- ( ( ph /\ A = B ) -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
7	2, 6	mpbird 247	. . 3 |- ( ( ph /\ A = B ) -> A e. { B , C } )
8	1, 7	mtand 691	. 2 |- ( ph -> -. A = B )
9	8	neqned 2801	1 |- ( ph -> A =/= B )

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:  ovnsubadd2lem  40859