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Theorem ndmovrcl 6443
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 6441 . . . 4  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  (/) )
43eleq1d 2536 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( ( A F B )  e.  S  <->  (/)  e.  S ) )
51, 4mtbiri 303 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  -.  ( A F B )  e.  S )
65con4i 130 1  |-  ( ( A F B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   (/)c0 3785    X. cxp 4997   dom cdm 4999  (class class class)co 6282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-iota 5549  df-fv 5594  df-ov 6285
This theorem is referenced by:  ndmovass  6445  ndmovdistr  6446  ndmovord  6447  ndmovordi  6448  caovmo  6494  brecop2  7402  eceqoveq  7413  addcanpi  9273  mulcanpi  9274  ordpipq  9316  recmulnq  9338  recclnq  9340  ltexnq  9349  nsmallnq  9351  ltbtwnnq  9352  prlem934  9407  ltaddpr  9408  ltaddpr2  9409  ltexprlem2  9411  ltexprlem3  9412  ltexprlem4  9413  ltexprlem6  9415  ltexprlem7  9416  addcanpr  9420  prlem936  9421  mappsrpr  9481
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