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Theorem ndmovordi 6027
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 4753 . . . 4  |-  ( ( C F A ) R ( C F B )  ->  (
( C F A )  e.  S  /\  ( C F B )  e.  S ) )
32simpld 445 . . 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 6022 . . . 4  |-  ( ( C F A )  e.  S  ->  ( C  e.  S  /\  A  e.  S )
)
76simpld 445 . . 3  |-  ( ( C F A )  e.  S  ->  C  e.  S )
83, 7syl 15 . 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 214 . 2  |-  ( C  e.  S  ->  (
( C F A ) R ( C F B )  ->  A R B ) )
118, 10mpcom 32 1  |-  ( ( C F A ) R ( C F B )  ->  A R B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696    C_ wss 3165   (/)c0 3468   class class class wbr 4039    X. cxp 4703   dom cdm 4705  (class class class)co 5874
This theorem is referenced by:  ltexprlem4  8679  ltsosr  8732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-iota 5235  df-fv 5279  df-ov 5877
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