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Theorem sdomdif 7009
Description: The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
Assertion
Ref Expression
sdomdif  |-  ( A 
~<  B  ->  ( B 
\  A )  =/=  (/) )

Proof of Theorem sdomdif
StepHypRef Expression
1 relsdom 6870 . . . . . 6  |-  Rel  ~<
21brrelexi 4729 . . . . 5  |-  ( A 
~<  B  ->  A  e. 
_V )
3 ssdif0 3513 . . . . . 6  |-  ( B 
C_  A  <->  ( B  \  A )  =  (/) )
4 ssdomg 6907 . . . . . . 7  |-  ( A  e.  _V  ->  ( B  C_  A  ->  B  ~<_  A ) )
5 domnsym 6987 . . . . . . 7  |-  ( B  ~<_  A  ->  -.  A  ~<  B )
64, 5syl6 29 . . . . . 6  |-  ( A  e.  _V  ->  ( B  C_  A  ->  -.  A  ~<  B ) )
73, 6syl5bir 209 . . . . 5  |-  ( A  e.  _V  ->  (
( B  \  A
)  =  (/)  ->  -.  A  ~<  B ) )
82, 7syl 15 . . . 4  |-  ( A 
~<  B  ->  ( ( B  \  A )  =  (/)  ->  -.  A  ~<  B ) )
98con2d 107 . . 3  |-  ( A 
~<  B  ->  ( A 
~<  B  ->  -.  ( B  \  A )  =  (/) ) )
109pm2.43i 43 . 2  |-  ( A 
~<  B  ->  -.  ( B  \  A )  =  (/) )
11 df-ne 2448 . 2  |-  ( ( B  \  A )  =/=  (/)  <->  -.  ( B  \  A )  =  (/) )
1210, 11sylibr 203 1  |-  ( A 
~<  B  ->  ( B 
\  A )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   class class class wbr 4023    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  domtriomlem  8068  konigthlem  8190  odcau  14915
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  ax-un 4512
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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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