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Theorem sdomdif 7193
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 7054 . . . . . 6  |-  Rel  ~<
21brrelexi 4860 . . . . 5  |-  ( A 
~<  B  ->  A  e. 
_V )
3 ssdif0 3631 . . . . . 6  |-  ( B 
C_  A  <->  ( B  \  A )  =  (/) )
4 ssdomg 7091 . . . . . . 7  |-  ( A  e.  _V  ->  ( B  C_  A  ->  B  ~<_  A ) )
5 domnsym 7171 . . . . . . 7  |-  ( B  ~<_  A  ->  -.  A  ~<  B )
64, 5syl6 31 . . . . . 6  |-  ( A  e.  _V  ->  ( B  C_  A  ->  -.  A  ~<  B ) )
73, 6syl5bir 210 . . . . 5  |-  ( A  e.  _V  ->  (
( B  \  A
)  =  (/)  ->  -.  A  ~<  B ) )
82, 7syl 16 . . . 4  |-  ( A 
~<  B  ->  ( ( B  \  A )  =  (/)  ->  -.  A  ~<  B ) )
98con2d 109 . . 3  |-  ( A 
~<  B  ->  ( A 
~<  B  ->  -.  ( B  \  A )  =  (/) ) )
109pm2.43i 45 . 2  |-  ( A 
~<  B  ->  -.  ( B  \  A )  =  (/) )
1110neneqad 2622 1  |-  ( A 
~<  B  ->  ( B 
\  A )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2552   _Vcvv 2901    \ cdif 3262    C_ wss 3265   (/)c0 3573   class class class wbr 4155    ~<_ cdom 7045    ~< csdm 7046
This theorem is referenced by:  domtriomlem  8257  konigthlem  8378  odcau  15167
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050
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