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Theorem ndmovordi 6238
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 4926 . . . 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 6233 . . . 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 1652    e. wcel 1725    C_ wss 3320   (/)c0 3628   class class class wbr 4212    X. cxp 4876   dom cdm 4878  (class class class)co 6081
This theorem is referenced by:  ltexprlem4  8916  ltsosr  8969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-dm 4888  df-iota 5418  df-fv 5462  df-ov 6084
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