Theorem dalemccnedd 32685

Description: Lemma for dath 32734. 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 1033	. . 3 |- ( ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) -> d =/= c )
3	1, 2	sylbi 195	. 2 |- ( ps -> d =/= c )
4	3	necomd 2674	1 |- ( ps -> c =/= d )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   e. wcel 1842   =/= wne 2598   class class class wbr 4394  (class class class)co 6234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-cleq 2394  df-ne 2600

This theorem is referenced by:  dalemswapyzps  32688  dalemrotps  32689  dalemcjden  32690