Theorem dif1card 8833

Description: The cardinality of a nonempty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
dif1card	|- ( ( A e. Fin /\ X e. A ) -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) )

Proof of Theorem dif1card

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		diffi 8192	. . 3 |- ( A e. Fin -> ( A \ { X } ) e. Fin )
2		isfi 7979	. . . 4 |- ( ( A \ { X } ) e. Fin <-> E. m e. om ( A \ { X } ) ~~ m )
3		simp3 1063	. . . . . . . . . . 11 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> ( A \ { X } ) ~~ m )
4		en2sn 8037	. . . . . . . . . . . 12 |- ( ( X e. A /\ m e. om ) -> { X } ~~ { m } )
5	4	3adant3 1081	. . . . . . . . . . 11 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> { X } ~~ { m } )
6		incom 3805	. . . . . . . . . . . . 13 |- ( ( A \ { X } ) i^i { X } ) = ( { X } i^i ( A \ { X } ) )
7		disjdif 4040	. . . . . . . . . . . . 13 |- ( { X } i^i ( A \ { X } ) ) = (/)
8	6, 7	eqtri 2644	. . . . . . . . . . . 12 |- ( ( A \ { X } ) i^i { X } ) = (/)
9	8	a1i 11	. . . . . . . . . . 11 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> ( ( A \ { X } ) i^i { X } ) = (/) )
10		nnord 7073	. . . . . . . . . . . . . 14 |- ( m e. om -> Ord m )
11		ordirr 5741	. . . . . . . . . . . . . 14 |- ( Ord m -> -. m e. m )
12	10, 11	syl 17	. . . . . . . . . . . . 13 |- ( m e. om -> -. m e. m )
13		disjsn 4246	. . . . . . . . . . . . 13 |- ( ( m i^i { m } ) = (/) <-> -. m e. m )
14	12, 13	sylibr 224	. . . . . . . . . . . 12 |- ( m e. om -> ( m i^i { m } ) = (/) )
15	14	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> ( m i^i { m } ) = (/) )
16		unen 8040	. . . . . . . . . . 11 |- ( ( ( ( A \ { X } ) ~~ m /\ { X } ~~ { m } ) /\ ( ( ( A \ { X } ) i^i { X } ) = (/) /\ ( m i^i { m } ) = (/) ) ) -> ( ( A \ { X } ) u. { X } ) ~~ ( m u. { m } ) )
17	3, 5, 9, 15, 16	syl22anc 1327	. . . . . . . . . 10 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> ( ( A \ { X } ) u. { X } ) ~~ ( m u. { m } ) )
18		difsnid 4341	. . . . . . . . . . . 12 |- ( X e. A -> ( ( A \ { X } ) u. { X } ) = A )
19		df-suc 5729	. . . . . . . . . . . . . 14 |- suc m = ( m u. { m } )
20	19	eqcomi 2631	. . . . . . . . . . . . 13 |- ( m u. { m } ) = suc m
21	20	a1i 11	. . . . . . . . . . . 12 |- ( X e. A -> ( m u. { m } ) = suc m )
22	18, 21	breq12d 4666	. . . . . . . . . . 11 |- ( X e. A -> ( ( ( A \ { X } ) u. { X } ) ~~ ( m u. { m } ) <-> A ~~ suc m ) )
23	22	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> ( ( ( A \ { X } ) u. { X } ) ~~ ( m u. { m } ) <-> A ~~ suc m ) )
24	17, 23	mpbid 222	. . . . . . . . 9 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> A ~~ suc m )
25		peano2 7086	. . . . . . . . . 10 |- ( m e. om -> suc m e. om )
26	25	3ad2ant2 1083	. . . . . . . . 9 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> suc m e. om )
27		cardennn 8809	. . . . . . . . 9 |- ( ( A ~~ suc m /\ suc m e. om ) -> ( card ` A ) = suc m )
28	24, 26, 27	syl2anc 693	. . . . . . . 8 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> ( card ` A ) = suc m )
29		cardennn 8809	. . . . . . . . . . 11 |- ( ( ( A \ { X } ) ~~ m /\ m e. om ) -> ( card ` ( A \ { X } ) ) = m )
30	29	ancoms 469	. . . . . . . . . 10 |- ( ( m e. om /\ ( A \ { X } ) ~~ m ) -> ( card ` ( A \ { X } ) ) = m )
31	30	3adant1 1079	. . . . . . . . 9 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> ( card ` ( A \ { X } ) ) = m )
32		suceq 5790	. . . . . . . . 9 |- ( ( card ` ( A \ { X } ) ) = m -> suc ( card ` ( A \ { X } ) ) = suc m )
33	31, 32	syl 17	. . . . . . . 8 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> suc ( card ` ( A \ { X } ) ) = suc m )
34	28, 33	eqtr4d 2659	. . . . . . 7 |- ( ( X e. A /\ m e. om /\ ( A \ { X } ) ~~ m ) -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) )
35	34	3expib 1268	. . . . . 6 |- ( X e. A -> ( ( m e. om /\ ( A \ { X } ) ~~ m ) -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) )
36	35	com12 32	. . . . 5 |- ( ( m e. om /\ ( A \ { X } ) ~~ m ) -> ( X e. A -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) )
37	36	rexlimiva 3028	. . . 4 |- ( E. m e. om ( A \ { X } ) ~~ m -> ( X e. A -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) )
38	2, 37	sylbi 207	. . 3 |- ( ( A \ { X } ) e. Fin -> ( X e. A -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) )
39	1, 38	syl 17	. 2 |- ( A e. Fin -> ( X e. A -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) )
40	39	imp 445	1 |- ( ( A e. Fin /\ X e. A ) -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   Ord word 5722   suc csuc 5725   ` cfv 5888   omcom 7065   ~~ cen 7952   Fincfn 7955   cardccrd 8761

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-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-int 4476  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-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by: (None)