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Theorem infdiffi 7374
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 3301 . . . . . 6  |-  ( x  =  (/)  ->  ( A 
\  x )  =  ( A  \  (/) ) )
2 dif0 3537 . . . . . 6  |-  ( A 
\  (/) )  =  A
31, 2syl6eq 2344 . . . . 5  |-  ( x  =  (/)  ->  ( A 
\  x )  =  A )
43breq1d 4049 . . . 4  |-  ( x  =  (/)  ->  ( ( A  \  x ) 
~~  A  <->  A  ~~  A ) )
54imbi2d 307 . . 3  |-  ( x  =  (/)  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A )  <->  ( om  ~<_  A  ->  A  ~~  A
) ) )
6 difeq2 3301 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
76breq1d 4049 . . . 4  |-  ( x  =  y  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  y )  ~~  A
) )
87imbi2d 307 . . 3  |-  ( x  =  y  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  y )  ~~  A ) ) )
9 difeq2 3301 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( A  \  ( y  u.  { z } ) ) )
10 difun1 3441 . . . . . 6  |-  ( A 
\  ( y  u. 
{ z } ) )  =  ( ( A  \  y ) 
\  { z } )
119, 10syl6eq 2344 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( ( A  \  y
)  \  { z } ) )
1211breq1d 4049 . . . 4  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( A 
\  x )  ~~  A 
<->  ( ( A  \ 
y )  \  {
z } )  ~~  A ) )
1312imbi2d 307 . . 3  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A
)  <->  ( om  ~<_  A  -> 
( ( A  \ 
y )  \  {
z } )  ~~  A ) ) )
14 difeq2 3301 . . . . 5  |-  ( x  =  B  ->  ( A  \  x )  =  ( A  \  B
) )
1514breq1d 4049 . . . 4  |-  ( x  =  B  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  B )  ~~  A
) )
1615imbi2d 307 . . 3  |-  ( x  =  B  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  B )  ~~  A ) ) )
17 reldom 6885 . . . . 5  |-  Rel  ~<_
1817brrelex2i 4746 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
19 enrefg 6909 . . . 4  |-  ( A  e.  _V  ->  A  ~~  A )
2018, 19syl 15 . . 3  |-  ( om  ~<_  A  ->  A  ~~  A )
21 domen2 7020 . . . . . . . . 9  |-  ( ( A  \  y ) 
~~  A  ->  ( om 
~<_  ( A  \  y
)  <->  om  ~<_  A ) )
2221biimparc 473 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  om  ~<_  ( A 
\  y ) )
23 infdifsn 7373 . . . . . . . 8  |-  ( om  ~<_  ( A  \  y
)  ->  ( ( A  \  y )  \  { z } ) 
~~  ( A  \ 
y ) )
2422, 23syl 15 . . . . . . 7  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  (
( A  \  y
)  \  { z } )  ~~  ( A  \  y ) )
25 entr 6929 . . . . . . 7  |-  ( ( ( ( A  \ 
y )  \  {
z } )  ~~  ( A  \  y
)  /\  ( A  \  y )  ~~  A
)  ->  ( ( A  \  y )  \  { z } ) 
~~  A )
2624, 25sylancom 648 . . . . . 6  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  (
( A  \  y
)  \  { z } )  ~~  A
)
2726ex 423 . . . . 5  |-  ( om  ~<_  A  ->  ( ( A  \  y )  ~~  A  ->  ( ( A 
\  y )  \  { z } ) 
~~  A ) )
2827a2i 12 . . . 4  |-  ( ( om  ~<_  A  ->  ( A  \  y )  ~~  A )  ->  ( om 
~<_  A  ->  ( ( A  \  y )  \  { z } ) 
~~  A ) )
2928a1i 10 . . 3  |-  ( y  e.  Fin  ->  (
( om  ~<_  A  -> 
( A  \  y
)  ~~  A )  ->  ( om  ~<_  A  -> 
( ( A  \ 
y )  \  {
z } )  ~~  A ) ) )
305, 8, 13, 16, 20, 29findcard2 7114 . 2  |-  ( B  e.  Fin  ->  ( om 
~<_  A  ->  ( A  \  B )  ~~  A
) )
3130impcom 419 1  |-  ( ( om  ~<_  A  /\  B  e.  Fin )  ->  ( A  \  B )  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163   (/)c0 3468   {csn 3653   class class class wbr 4039   omcom 4672    ~~ cen 6876    ~<_ cdom 6877   Fincfn 6879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-fin 6883
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