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Theorem ndmovordi 6197
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
StepHypRef Expression
1 ndmovordi.4 . . . . 5  |-  R  C_  ( S  X.  S
)
21brel 4885 . . . 4  |-  ( ( C F A ) R ( C F B )  ->  (
( C F A )  e.  S  /\  ( C F B )  e.  S ) )
32simpld 446 . . 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
64, 5ndmovrcl 6192 . . . 4  |-  ( ( C F A )  e.  S  ->  ( C  e.  S  /\  A  e.  S )
)
76simpld 446 . . 3  |-  ( ( C F A )  e.  S  ->  C  e.  S )
83, 7syl 16 . 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 ) ) )
109biimprd 215 . 2  |-  ( C  e.  S  ->  (
( C F A ) R ( C F B )  ->  A R B ) )
118, 10mpcom 34 1  |-  ( ( C F A ) R ( C F B )  ->  A R B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721    C_ wss 3280   (/)c0 3588   class class class wbr 4172    X. cxp 4835   dom cdm 4837  (class class class)co 6040
This theorem is referenced by:  ltexprlem4  8872  ltsosr  8925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-dm 4847  df-iota 5377  df-fv 5421  df-ov 6043
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