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Theorem infdiffi 8106
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 3554 . . . . . 6  |-  ( x  =  (/)  ->  ( A 
\  x )  =  ( A  \  (/) ) )
2 dif0 3841 . . . . . 6  |-  ( A 
\  (/) )  =  A
31, 2syl6eq 2459 . . . . 5  |-  ( x  =  (/)  ->  ( A 
\  x )  =  A )
43breq1d 4404 . . . 4  |-  ( x  =  (/)  ->  ( ( A  \  x ) 
~~  A  <->  A  ~~  A ) )
54imbi2d 314 . . 3  |-  ( x  =  (/)  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A )  <->  ( om  ~<_  A  ->  A  ~~  A
) ) )
6 difeq2 3554 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
76breq1d 4404 . . . 4  |-  ( x  =  y  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  y )  ~~  A
) )
87imbi2d 314 . . 3  |-  ( x  =  y  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  y )  ~~  A ) ) )
9 difeq2 3554 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( A  \  ( y  u.  { z } ) ) )
10 difun1 3709 . . . . . 6  |-  ( A 
\  ( y  u. 
{ z } ) )  =  ( ( A  \  y ) 
\  { z } )
119, 10syl6eq 2459 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( ( A  \  y
)  \  { z } ) )
1211breq1d 4404 . . . 4  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( A 
\  x )  ~~  A 
<->  ( ( A  \ 
y )  \  {
z } )  ~~  A ) )
1312imbi2d 314 . . 3  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A
)  <->  ( om  ~<_  A  -> 
( ( A  \ 
y )  \  {
z } )  ~~  A ) ) )
14 difeq2 3554 . . . . 5  |-  ( x  =  B  ->  ( A  \  x )  =  ( A  \  B
) )
1514breq1d 4404 . . . 4  |-  ( x  =  B  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  B )  ~~  A
) )
1615imbi2d 314 . . 3  |-  ( x  =  B  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  B )  ~~  A ) ) )
17 reldom 7559 . . . . 5  |-  Rel  ~<_
1817brrelex2i 4864 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
19 enrefg 7584 . . . 4  |-  ( A  e.  _V  ->  A  ~~  A )
2018, 19syl 17 . . 3  |-  ( om  ~<_  A  ->  A  ~~  A )
21 domen2 7697 . . . . . . . . 9  |-  ( ( A  \  y ) 
~~  A  ->  ( om 
~<_  ( A  \  y
)  <->  om  ~<_  A ) )
2221biimparc 485 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  om  ~<_  ( A 
\  y ) )
23 infdifsn 8105 . . . . . . . 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 7604 . . . . . . 7  |-  ( ( ( ( A  \ 
y )  \  {
z } )  ~~  ( A  \  y
)  /\  ( A  \  y )  ~~  A
)  ->  ( ( A  \  y )  \  { z } ) 
~~  A )
2624, 25sylancom 665 . . . . . 6  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  (
( A  \  y
)  \  { z } )  ~~  A
)
2726ex 432 . . . . 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 7793 . 2  |-  ( B  e.  Fin  ->  ( om 
~<_  A  ->  ( A  \  B )  ~~  A
) )
3130impcom 428 1  |-  ( ( om  ~<_  A  /\  B  e.  Fin )  ->  ( A  \  B )  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    \ cdif 3410    u. cun 3411   (/)c0 3737   {csn 3971   class class class wbr 4394   omcom 6682    ~~ cen 7550    ~<_ cdom 7551   Fincfn 7553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-1o 7166  df-er 7347  df-en 7554  df-dom 7555  df-fin 7557
This theorem is referenced by: (None)
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