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Theorem ndmovrcl 6006
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 6004 . . . 4  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  (/) )
43eleq1d 2349 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( ( A F B )  e.  S  <->  (/)  e.  S ) )
51, 4mtbiri 294 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  -.  ( A F B )  e.  S )
65con4i 122 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 358    = wceq 1623    e. wcel 1684   (/)c0 3455    X. cxp 4687   dom cdm 4689  (class class class)co 5858
This theorem is referenced by:  ndmovass  6008  ndmovdistr  6009  ndmovord  6010  ndmovordi  6011  caovmo  6057  brecop2  6752  eceqoveq  6763  addcanpi  8523  mulcanpi  8524  ordpipq  8566  recmulnq  8588  recclnq  8590  ltexnq  8599  nsmallnq  8601  ltbtwnnq  8602  prlem934  8657  ltaddpr  8658  ltaddpr2  8659  ltexprlem2  8661  ltexprlem3  8662  ltexprlem4  8663  ltexprlem6  8665  ltexprlem7  8666  addcanpr  8670  prlem936  8671  mappsrpr  8730
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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