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Theorem ndmovord 6010
Description: Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)
Hypotheses
Ref Expression
ndmov.1  |-  dom  F  =  ( S  X.  S )
ndmovord.4  |-  R  C_  ( S  X.  S
)
ndmovord.5  |-  -.  (/)  e.  S
ndmovord.6  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A R B  <-> 
( C F A ) R ( C F B ) ) )
Assertion
Ref Expression
ndmovord  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )

Proof of Theorem ndmovord
StepHypRef Expression
1 ndmovord.6 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A R B  <-> 
( C F A ) R ( C F B ) ) )
213expia 1153 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( C  e.  S  ->  ( A R B  <-> 
( C F A ) R ( C F B ) ) ) )
3 ndmovord.4 . . . . 5  |-  R  C_  ( S  X.  S
)
43brel 4737 . . . 4  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
53brel 4737 . . . . 5  |-  ( ( C F A ) R ( C F B )  ->  (
( C F A )  e.  S  /\  ( C F B )  e.  S ) )
6 ndmov.1 . . . . . . . 8  |-  dom  F  =  ( S  X.  S )
7 ndmovord.5 . . . . . . . 8  |-  -.  (/)  e.  S
86, 7ndmovrcl 6006 . . . . . . 7  |-  ( ( C F A )  e.  S  ->  ( C  e.  S  /\  A  e.  S )
)
98simprd 449 . . . . . 6  |-  ( ( C F A )  e.  S  ->  A  e.  S )
106, 7ndmovrcl 6006 . . . . . . 7  |-  ( ( C F B )  e.  S  ->  ( C  e.  S  /\  B  e.  S )
)
1110simprd 449 . . . . . 6  |-  ( ( C F B )  e.  S  ->  B  e.  S )
129, 11anim12i 549 . . . . 5  |-  ( ( ( C F A )  e.  S  /\  ( C F B )  e.  S )  -> 
( A  e.  S  /\  B  e.  S
) )
135, 12syl 15 . . . 4  |-  ( ( C F A ) R ( C F B )  ->  ( A  e.  S  /\  B  e.  S )
)
144, 13pm5.21ni 341 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
1514a1d 22 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) ) )
162, 15pm2.61i 156 1  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   class class class wbr 4023    X. cxp 4687   dom cdm 4689  (class class class)co 5858
This theorem is referenced by:  ltapi  8527  ltmpi  8528  ltanq  8595  ltmnq  8596  ltapr  8669  ltasr  8722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861
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