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Theorem infdiffi 7604
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 3451 . . . . . 6  |-  ( x  =  (/)  ->  ( A 
\  x )  =  ( A  \  (/) ) )
2 dif0 3690 . . . . . 6  |-  ( A 
\  (/) )  =  A
31, 2syl6eq 2483 . . . . 5  |-  ( x  =  (/)  ->  ( A 
\  x )  =  A )
43breq1d 4214 . . . 4  |-  ( x  =  (/)  ->  ( ( A  \  x ) 
~~  A  <->  A  ~~  A ) )
54imbi2d 308 . . 3  |-  ( x  =  (/)  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A )  <->  ( om  ~<_  A  ->  A  ~~  A
) ) )
6 difeq2 3451 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
76breq1d 4214 . . . 4  |-  ( x  =  y  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  y )  ~~  A
) )
87imbi2d 308 . . 3  |-  ( x  =  y  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  y )  ~~  A ) ) )
9 difeq2 3451 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( A  \  ( y  u.  { z } ) ) )
10 difun1 3593 . . . . . 6  |-  ( A 
\  ( y  u. 
{ z } ) )  =  ( ( A  \  y ) 
\  { z } )
119, 10syl6eq 2483 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( ( A  \  y
)  \  { z } ) )
1211breq1d 4214 . . . 4  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( A 
\  x )  ~~  A 
<->  ( ( A  \ 
y )  \  {
z } )  ~~  A ) )
1312imbi2d 308 . . 3  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A
)  <->  ( om  ~<_  A  -> 
( ( A  \ 
y )  \  {
z } )  ~~  A ) ) )
14 difeq2 3451 . . . . 5  |-  ( x  =  B  ->  ( A  \  x )  =  ( A  \  B
) )
1514breq1d 4214 . . . 4  |-  ( x  =  B  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  B )  ~~  A
) )
1615imbi2d 308 . . 3  |-  ( x  =  B  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  B )  ~~  A ) ) )
17 reldom 7107 . . . . 5  |-  Rel  ~<_
1817brrelex2i 4911 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
19 enrefg 7131 . . . 4  |-  ( A  e.  _V  ->  A  ~~  A )
2018, 19syl 16 . . 3  |-  ( om  ~<_  A  ->  A  ~~  A )
21 domen2 7242 . . . . . . . . 9  |-  ( ( A  \  y ) 
~~  A  ->  ( om 
~<_  ( A  \  y
)  <->  om  ~<_  A ) )
2221biimparc 474 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  om  ~<_  ( A 
\  y ) )
23 infdifsn 7603 . . . . . . . 8  |-  ( om  ~<_  ( A  \  y
)  ->  ( ( A  \  y )  \  { z } ) 
~~  ( A  \ 
y ) )
2422, 23syl 16 . . . . . . 7  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  (
( A  \  y
)  \  { z } )  ~~  ( A  \  y ) )
25 entr 7151 . . . . . . 7  |-  ( ( ( ( A  \ 
y )  \  {
z } )  ~~  ( A  \  y
)  /\  ( A  \  y )  ~~  A
)  ->  ( ( A  \  y )  \  { z } ) 
~~  A )
2624, 25sylancom 649 . . . . . 6  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  (
( A  \  y
)  \  { z } )  ~~  A
)
2726ex 424 . . . . 5  |-  ( om  ~<_  A  ->  ( ( A  \  y )  ~~  A  ->  ( ( A 
\  y )  \  { z } ) 
~~  A ) )
2827a2i 13 . . . 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 7340 . 2  |-  ( B  e.  Fin  ->  ( om 
~<_  A  ->  ( A  \  B )  ~~  A
) )
3130impcom 420 1  |-  ( ( om  ~<_  A  /\  B  e.  Fin )  ->  ( A  \  B )  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310   (/)c0 3620   {csn 3806   class class class wbr 4204   omcom 4837    ~~ cen 7098    ~<_ cdom 7099   Fincfn 7101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-fin 7105
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