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Theorem sdomdif 7662
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 7520 . . . . . 6  |-  Rel  ~<
21brrelexi 5039 . . . . 5  |-  ( A 
~<  B  ->  A  e. 
_V )
3 ssdif0 3885 . . . . . 6  |-  ( B 
C_  A  <->  ( B  \  A )  =  (/) )
4 ssdomg 7558 . . . . . . 7  |-  ( A  e.  _V  ->  ( B  C_  A  ->  B  ~<_  A ) )
5 domnsym 7640 . . . . . . 7  |-  ( B  ~<_  A  ->  -.  A  ~<  B )
64, 5syl6 33 . . . . . 6  |-  ( A  e.  _V  ->  ( B  C_  A  ->  -.  A  ~<  B ) )
73, 6syl5bir 218 . . . . 5  |-  ( A  e.  _V  ->  (
( B  \  A
)  =  (/)  ->  -.  A  ~<  B ) )
82, 7syl 16 . . . 4  |-  ( A 
~<  B  ->  ( ( B  \  A )  =  (/)  ->  -.  A  ~<  B ) )
98con2d 115 . . 3  |-  ( A 
~<  B  ->  ( A 
~<  B  ->  -.  ( B  \  A )  =  (/) ) )
109pm2.43i 47 . 2  |-  ( A 
~<  B  ->  -.  ( B  \  A )  =  (/) )
1110neqned 2670 1  |-  ( A 
~<  B  ->  ( B 
\  A )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   class class class wbr 4447    ~<_ cdom 7511    ~< csdm 7512
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  ax-un 6574
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-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516
This theorem is referenced by:  domtriomlem  8818  konigthlem  8939  odcau  16420
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