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Theorem ndmovrcl 6196
Description: Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.)
Hypotheses
Ref Expression
ndmov.1  |-  dom  F  =  ( S  X.  S )
ndmovrcl.3  |-  -.  (/)  e.  S
Assertion
Ref Expression
ndmovrcl  |-  ( ( A F B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)

Proof of Theorem ndmovrcl
StepHypRef Expression
1 ndmovrcl.3 . . 3  |-  -.  (/)  e.  S
2 ndmov.1 . . . . 5  |-  dom  F  =  ( S  X.  S )
32ndmov 6194 . . . 4  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  (/) )
43eleq1d 2474 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( ( A F B )  e.  S  <->  (/)  e.  S ) )
51, 4mtbiri 295 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  -.  ( A F B )  e.  S )
65con4i 124 1  |-  ( ( A F B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   (/)c0 3592    X. cxp 4839   dom cdm 4841  (class class class)co 6044
This theorem is referenced by:  ndmovass  6198  ndmovdistr  6199  ndmovord  6200  ndmovordi  6201  caovmo  6247  brecop2  6961  eceqoveq  6972  addcanpi  8736  mulcanpi  8737  ordpipq  8779  recmulnq  8801  recclnq  8803  ltexnq  8812  nsmallnq  8814  ltbtwnnq  8815  prlem934  8870  ltaddpr  8871  ltaddpr2  8872  ltexprlem2  8874  ltexprlem3  8875  ltexprlem4  8876  ltexprlem6  8878  ltexprlem7  8879  addcanpr  8883  prlem936  8884  mappsrpr  8943
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-xp 4847  df-dm 4851  df-iota 5381  df-fv 5425  df-ov 6047
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