Theorem ndmovordi 6825

Description: Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)

Hypotheses

Ref	Expression
ndmovordi.2	|- dom F = ( S X. S )
ndmovordi.4	|- R C_ ( S X. S )
ndmovordi.5	|- -. (/) e. S
ndmovordi.6	|- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Assertion

Ref	Expression
ndmovordi	|- ( ( C F A ) R ( C F B ) -> A R B )

Proof of Theorem ndmovordi

Step	Hyp	Ref	Expression
1		ndmovordi.4	. . . . 5 |- R C_ ( S X. S )
2	1	brel 5168	. . . 4 |- ( ( C F A ) R ( C F B ) -> ( ( C F A ) e. S /\ ( C F B ) e. S ) )
3	2	simpld 475	. . 3 |- ( ( C F A ) R ( C F B ) -> ( C F A ) e. S )
4		ndmovordi.2	. . . . 5 |- dom F = ( S X. S )
5		ndmovordi.5	. . . . 5 |- -. (/) e. S
6	4, 5	ndmovrcl 6820	. . . 4 |- ( ( C F A ) e. S -> ( C e. S /\ A e. S ) )
7	6	simpld 475	. . 3 |- ( ( C F A ) e. S -> C e. S )
8	3, 7	syl 17	. 2 |- ( ( C F A ) R ( C F B ) -> C e. S )
9		ndmovordi.6	. . 3 |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )
10	9	biimprd 238	. 2 |- ( C e. S -> ( ( C F A ) R ( C F B ) -> A R B ) )
11	8, 10	mpcom 38	1 |- ( ( C F A ) R ( C F B ) -> A R B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   C_ wss 3574   (/)c0 3915   class class class wbr 4653   X. cxp 5112   dom cdm 5114  (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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  ltexprlem4  9861  ltsosr  9915