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Theorem infdiffi 8171
Description: Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
infdiffi  |-  ( ( om  ~<_  A  /\  B  e.  Fin )  ->  ( A  \  B )  ~~  A )

Proof of Theorem infdiffi
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difeq2 3577 . . . . . 6  |-  ( x  =  (/)  ->  ( A 
\  x )  =  ( A  \  (/) ) )
2 dif0 3867 . . . . . 6  |-  ( A 
\  (/) )  =  A
31, 2syl6eq 2479 . . . . 5  |-  ( x  =  (/)  ->  ( A 
\  x )  =  A )
43breq1d 4433 . . . 4  |-  ( x  =  (/)  ->  ( ( A  \  x ) 
~~  A  <->  A  ~~  A ) )
54imbi2d 317 . . 3  |-  ( x  =  (/)  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A )  <->  ( om  ~<_  A  ->  A  ~~  A
) ) )
6 difeq2 3577 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
76breq1d 4433 . . . 4  |-  ( x  =  y  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  y )  ~~  A
) )
87imbi2d 317 . . 3  |-  ( x  =  y  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  y )  ~~  A ) ) )
9 difeq2 3577 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( A  \  ( y  u.  { z } ) ) )
10 difun1 3733 . . . . . 6  |-  ( A 
\  ( y  u. 
{ z } ) )  =  ( ( A  \  y ) 
\  { z } )
119, 10syl6eq 2479 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( ( A  \  y
)  \  { z } ) )
1211breq1d 4433 . . . 4  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( A 
\  x )  ~~  A 
<->  ( ( A  \ 
y )  \  {
z } )  ~~  A ) )
1312imbi2d 317 . . 3  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A
)  <->  ( om  ~<_  A  -> 
( ( A  \ 
y )  \  {
z } )  ~~  A ) ) )
14 difeq2 3577 . . . . 5  |-  ( x  =  B  ->  ( A  \  x )  =  ( A  \  B
) )
1514breq1d 4433 . . . 4  |-  ( x  =  B  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  B )  ~~  A
) )
1615imbi2d 317 . . 3  |-  ( x  =  B  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  B )  ~~  A ) ) )
17 reldom 7586 . . . . 5  |-  Rel  ~<_
1817brrelex2i 4895 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
19 enrefg 7611 . . . 4  |-  ( A  e.  _V  ->  A  ~~  A )
2018, 19syl 17 . . 3  |-  ( om  ~<_  A  ->  A  ~~  A )
21 domen2 7724 . . . . . . . . 9  |-  ( ( A  \  y ) 
~~  A  ->  ( om 
~<_  ( A  \  y
)  <->  om  ~<_  A ) )
2221biimparc 489 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  om  ~<_  ( A 
\  y ) )
23 infdifsn 8170 . . . . . . . 8  |-  ( om  ~<_  ( A  \  y
)  ->  ( ( A  \  y )  \  { z } ) 
~~  ( A  \ 
y ) )
2422, 23syl 17 . . . . . . 7  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  (
( A  \  y
)  \  { z } )  ~~  ( A  \  y ) )
25 entr 7631 . . . . . . 7  |-  ( ( ( ( A  \ 
y )  \  {
z } )  ~~  ( A  \  y
)  /\  ( A  \  y )  ~~  A
)  ->  ( ( A  \  y )  \  { z } ) 
~~  A )
2624, 25sylancom 671 . . . . . 6  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  (
( A  \  y
)  \  { z } )  ~~  A
)
2726ex 435 . . . . 5  |-  ( om  ~<_  A  ->  ( ( A  \  y )  ~~  A  ->  ( ( A 
\  y )  \  { z } ) 
~~  A ) )
2827a2i 14 . . . 4  |-  ( ( om  ~<_  A  ->  ( A  \  y )  ~~  A )  ->  ( om 
~<_  A  ->  ( ( A  \  y )  \  { z } ) 
~~  A ) )
2928a1i 11 . . 3  |-  ( y  e.  Fin  ->  (
( om  ~<_  A  -> 
( A  \  y
)  ~~  A )  ->  ( om  ~<_  A  -> 
( ( A  \ 
y )  \  {
z } )  ~~  A ) ) )
305, 8, 13, 16, 20, 29findcard2 7820 . 2  |-  ( B  e.  Fin  ->  ( om 
~<_  A  ->  ( A  \  B )  ~~  A
) )
3130impcom 431 1  |-  ( ( om  ~<_  A  /\  B  e.  Fin )  ->  ( A  \  B )  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080    \ cdif 3433    u. cun 3434   (/)c0 3761   {csn 3998   class class class wbr 4423   omcom 6706    ~~ cen 7577    ~<_ cdom 7578   Fincfn 7580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-1o 7193  df-er 7374  df-en 7581  df-dom 7582  df-fin 7584
This theorem is referenced by: (None)
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