Theorem dalemccnedd 34973

Description: Lemma for dath 35022. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Hypothesis

Ref	Expression
da.ps0	|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )

Assertion

Ref	Expression
dalemccnedd	|- ( ps -> c =/= d )

Proof of Theorem dalemccnedd

Step	Hyp	Ref	Expression
1		da.ps0	. . 3 |- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
2		simp31 1097	. . 3 |- ( ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) -> d =/= c )
3	1, 2	sylbi 207	. 2 |- ( ps -> d =/= c )
4	3	necomd 2849	1 |- ( ps -> c =/= d )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650

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-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705  df-cleq 2615  df-ne 2795

This theorem is referenced by:  dalemswapyzps  34976  dalemrotps  34977  dalemcjden  34978