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Theorem sdomdif 7664
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 7522 . . . . . 6  |-  Rel  ~<
21brrelexi 5027 . . . . 5  |-  ( A 
~<  B  ->  A  e. 
_V )
3 ssdif0 3868 . . . . . 6  |-  ( B 
C_  A  <->  ( B  \  A )  =  (/) )
4 ssdomg 7560 . . . . . . 7  |-  ( A  e.  _V  ->  ( B  C_  A  ->  B  ~<_  A ) )
5 domnsym 7642 . . . . . . 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 2644 1  |-  ( A 
~<  B  ->  ( B 
\  A )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1381    e. wcel 1802    =/= wne 2636   _Vcvv 3093    \ cdif 3456    C_ wss 3459   (/)c0 3768   class class class wbr 4434    ~<_ cdom 7513    ~< csdm 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518
This theorem is referenced by:  domtriomlem  8822  konigthlem  8943  odcau  16495
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