Theorem infdiffi 8555

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.

Step	Hyp	Ref	Expression
1		difeq2 3722	. . . . . 6 |- ( x = (/) -> ( A \ x ) = ( A \ (/) ) )
2		dif0 3950	. . . . . 6 |- ( A \ (/) ) = A
3	1, 2	syl6eq 2672	. . . . 5 |- ( x = (/) -> ( A \ x ) = A )
4	3	breq1d 4663	. . . 4 |- ( x = (/) -> ( ( A \ x ) ~~ A <-> A ~~ A ) )
5	4	imbi2d 330	. . 3 |- ( x = (/) -> ( ( om ~<_ A -> ( A \ x ) ~~ A ) <-> ( om ~<_ A -> A ~~ A ) ) )
6		difeq2 3722	. . . . 5 |- ( x = y -> ( A \ x ) = ( A \ y ) )
7	6	breq1d 4663	. . . 4 |- ( x = y -> ( ( A \ x ) ~~ A <-> ( A \ y ) ~~ A ) )
8	7	imbi2d 330	. . 3 |- ( x = y -> ( ( om ~<_ A -> ( A \ x ) ~~ A ) <-> ( om ~<_ A -> ( A \ y ) ~~ A ) ) )
9		difeq2 3722	. . . . . 6 |- ( x = ( y u. { z } ) -> ( A \ x ) = ( A \ ( y u. { z } ) ) )
10		difun1 3887	. . . . . 6 |- ( A \ ( y u. { z } ) ) = ( ( A \ y ) \ { z } )
11	9, 10	syl6eq 2672	. . . . 5 |- ( x = ( y u. { z } ) -> ( A \ x ) = ( ( A \ y ) \ { z } ) )
12	11	breq1d 4663	. . . 4 |- ( x = ( y u. { z } ) -> ( ( A \ x ) ~~ A <-> ( ( A \ y ) \ { z } ) ~~ A ) )
13	12	imbi2d 330	. . 3 |- ( x = ( y u. { z } ) -> ( ( om ~<_ A -> ( A \ x ) ~~ A ) <-> ( om ~<_ A -> ( ( A \ y ) \ { z } ) ~~ A ) ) )
14		difeq2 3722	. . . . 5 |- ( x = B -> ( A \ x ) = ( A \ B ) )
15	14	breq1d 4663	. . . 4 |- ( x = B -> ( ( A \ x ) ~~ A <-> ( A \ B ) ~~ A ) )
16	15	imbi2d 330	. . 3 |- ( x = B -> ( ( om ~<_ A -> ( A \ x ) ~~ A ) <-> ( om ~<_ A -> ( A \ B ) ~~ A ) ) )
17		reldom 7961	. . . . 5 |- Rel ~<_
18	17	brrelex2i 5159	. . . 4 |- ( om ~<_ A -> A e. _V )
19		enrefg 7987	. . . 4 |- ( A e. _V -> A ~~ A )
20	18, 19	syl 17	. . 3 |- ( om ~<_ A -> A ~~ A )
21		domen2 8103	. . . . . . . . 9 |- ( ( A \ y ) ~~ A -> ( om ~<_ ( A \ y ) <-> om ~<_ A ) )
22	21	biimparc 504	. . . . . . . 8 |- ( ( om ~<_ A /\ ( A \ y ) ~~ A ) -> om ~<_ ( A \ y ) )
23		infdifsn 8554	. . . . . . . 8 |- ( om ~<_ ( A \ y ) -> ( ( A \ y ) \ { z } ) ~~ ( A \ y ) )
24	22, 23	syl 17	. . . . . . 7 |- ( ( om ~<_ A /\ ( A \ y ) ~~ A ) -> ( ( A \ y ) \ { z } ) ~~ ( A \ y ) )
25		entr 8008	. . . . . . 7 |- ( ( ( ( A \ y ) \ { z } ) ~~ ( A \ y ) /\ ( A \ y ) ~~ A ) -> ( ( A \ y ) \ { z } ) ~~ A )
26	24, 25	sylancom 701	. . . . . 6 |- ( ( om ~<_ A /\ ( A \ y ) ~~ A ) -> ( ( A \ y ) \ { z } ) ~~ A )
27	26	ex 450	. . . . 5 |- ( om ~<_ A -> ( ( A \ y ) ~~ A -> ( ( A \ y ) \ { z } ) ~~ A ) )
28	27	a2i 14	. . . 4 |- ( ( om ~<_ A -> ( A \ y ) ~~ A ) -> ( om ~<_ A -> ( ( A \ y ) \ { z } ) ~~ A ) )
29	28	a1i 11	. . 3 |- ( y e. Fin -> ( ( om ~<_ A -> ( A \ y ) ~~ A ) -> ( om ~<_ A -> ( ( A \ y ) \ { z } ) ~~ A ) ) )
30	5, 8, 13, 16, 20, 29	findcard2 8200	. 2 |- ( B e. Fin -> ( om ~<_ A -> ( A \ B ) ~~ A ) )
31	30	impcom 446	1 |- ( ( om ~<_ A /\ B e. Fin ) -> ( A \ B ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   (/)c0 3915   {csn 4177   class class class wbr 4653   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959

This theorem is referenced by: (None)