Theorem infdif 9031

Description: The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
infdif	|- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A \ B ) ~~ A )

Proof of Theorem infdif

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> A e. dom card )
2		difss 3737	. . 3 |- ( A \ B ) C_ A
3		ssdomg 8001	. . 3 |- ( A e. dom card -> ( ( A \ B ) C_ A -> ( A \ B ) ~<_ A ) )
4	1, 2, 3	mpisyl 21	. 2 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A \ B ) ~<_ A )
5		sdomdom 7983	. . . . . . . . 9 |- ( B ~< A -> B ~<_ A )
6	5	3ad2ant3 1084	. . . . . . . 8 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> B ~<_ A )
7		numdom 8861	. . . . . . . 8 |- ( ( A e. dom card /\ B ~<_ A ) -> B e. dom card )
8	1, 6, 7	syl2anc 693	. . . . . . 7 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> B e. dom card )
9		unnum 9022	. . . . . . 7 |- ( ( A e. dom card /\ B e. dom card ) -> ( A u. B ) e. dom card )
10	1, 8, 9	syl2anc 693	. . . . . 6 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A u. B ) e. dom card )
11		ssun1 3776	. . . . . 6 |- A C_ ( A u. B )
12		ssdomg 8001	. . . . . 6 |- ( ( A u. B ) e. dom card -> ( A C_ ( A u. B ) -> A ~<_ ( A u. B ) ) )
13	10, 11, 12	mpisyl 21	. . . . 5 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> A ~<_ ( A u. B ) )
14		undif1 4043	. . . . . 6 |- ( ( A \ B ) u. B ) = ( A u. B )
15		ssnum 8862	. . . . . . . 8 |- ( ( A e. dom card /\ ( A \ B ) C_ A ) -> ( A \ B ) e. dom card )
16	1, 2, 15	sylancl 694	. . . . . . 7 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A \ B ) e. dom card )
17		uncdadom 8993	. . . . . . 7 |- ( ( ( A \ B ) e. dom card /\ B e. dom card ) -> ( ( A \ B ) u. B ) ~<_ ( ( A \ B ) +c B ) )
18	16, 8, 17	syl2anc 693	. . . . . 6 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( ( A \ B ) u. B ) ~<_ ( ( A \ B ) +c B ) )
19	14, 18	syl5eqbrr 4689	. . . . 5 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A u. B ) ~<_ ( ( A \ B ) +c B ) )
20		domtr 8009	. . . . 5 |- ( ( A ~<_ ( A u. B ) /\ ( A u. B ) ~<_ ( ( A \ B ) +c B ) ) -> A ~<_ ( ( A \ B ) +c B ) )
21	13, 19, 20	syl2anc 693	. . . 4 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> A ~<_ ( ( A \ B ) +c B ) )
22		simp3 1063	. . . . . . 7 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> B ~< A )
23		sdomdom 7983	. . . . . . . . 9 |- ( ( A \ B ) ~< B -> ( A \ B ) ~<_ B )
24		cdadom1 9008	. . . . . . . . 9 |- ( ( A \ B ) ~<_ B -> ( ( A \ B ) +c B ) ~<_ ( B +c B ) )
25	23, 24	syl 17	. . . . . . . 8 |- ( ( A \ B ) ~< B -> ( ( A \ B ) +c B ) ~<_ ( B +c B ) )
26		domtr 8009	. . . . . . . . . . 11 |- ( ( A ~<_ ( ( A \ B ) +c B ) /\ ( ( A \ B ) +c B ) ~<_ ( B +c B ) ) -> A ~<_ ( B +c B ) )
27	26	ex 450	. . . . . . . . . 10 |- ( A ~<_ ( ( A \ B ) +c B ) -> ( ( ( A \ B ) +c B ) ~<_ ( B +c B ) -> A ~<_ ( B +c B ) ) )
28	21, 27	syl 17	. . . . . . . . 9 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( ( ( A \ B ) +c B ) ~<_ ( B +c B ) -> A ~<_ ( B +c B ) ) )
29		simp2 1062	. . . . . . . . . . . 12 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> om ~<_ A )
30		domtr 8009	. . . . . . . . . . . . 13 |- ( ( om ~<_ A /\ A ~<_ ( B +c B ) ) -> om ~<_ ( B +c B ) )
31	30	ex 450	. . . . . . . . . . . 12 |- ( om ~<_ A -> ( A ~<_ ( B +c B ) -> om ~<_ ( B +c B ) ) )
32	29, 31	syl 17	. . . . . . . . . . 11 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A ~<_ ( B +c B ) -> om ~<_ ( B +c B ) ) )
33		cdainf 9014	. . . . . . . . . . . . 13 |- ( om ~<_ B <-> om ~<_ ( B +c B ) )
34	33	biimpri 218	. . . . . . . . . . . 12 |- ( om ~<_ ( B +c B ) -> om ~<_ B )
35		domrefg 7990	. . . . . . . . . . . . 13 |- ( B e. dom card -> B ~<_ B )
36		infcdaabs 9028	. . . . . . . . . . . . . . 15 |- ( ( B e. dom card /\ om ~<_ B /\ B ~<_ B ) -> ( B +c B ) ~~ B )
37	36	3com23 1271	. . . . . . . . . . . . . 14 |- ( ( B e. dom card /\ B ~<_ B /\ om ~<_ B ) -> ( B +c B ) ~~ B )
38	37	3expia 1267	. . . . . . . . . . . . 13 |- ( ( B e. dom card /\ B ~<_ B ) -> ( om ~<_ B -> ( B +c B ) ~~ B ) )
39	35, 38	mpdan 702	. . . . . . . . . . . 12 |- ( B e. dom card -> ( om ~<_ B -> ( B +c B ) ~~ B ) )
40	8, 34, 39	syl2im 40	. . . . . . . . . . 11 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( om ~<_ ( B +c B ) -> ( B +c B ) ~~ B ) )
41	32, 40	syld 47	. . . . . . . . . 10 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A ~<_ ( B +c B ) -> ( B +c B ) ~~ B ) )
42		domen2 8103	. . . . . . . . . . 11 |- ( ( B +c B ) ~~ B -> ( A ~<_ ( B +c B ) <-> A ~<_ B ) )
43	42	biimpcd 239	. . . . . . . . . 10 |- ( A ~<_ ( B +c B ) -> ( ( B +c B ) ~~ B -> A ~<_ B ) )
44	41, 43	sylcom 30	. . . . . . . . 9 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A ~<_ ( B +c B ) -> A ~<_ B ) )
45	28, 44	syld 47	. . . . . . . 8 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( ( ( A \ B ) +c B ) ~<_ ( B +c B ) -> A ~<_ B ) )
46		domnsym 8086	. . . . . . . 8 |- ( A ~<_ B -> -. B ~< A )
47	25, 45, 46	syl56 36	. . . . . . 7 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( ( A \ B ) ~< B -> -. B ~< A ) )
48	22, 47	mt2d 131	. . . . . 6 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> -. ( A \ B ) ~< B )
49		domtri2 8815	. . . . . . 7 |- ( ( B e. dom card /\ ( A \ B ) e. dom card ) -> ( B ~<_ ( A \ B ) <-> -. ( A \ B ) ~< B ) )
50	8, 16, 49	syl2anc 693	. . . . . 6 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( B ~<_ ( A \ B ) <-> -. ( A \ B ) ~< B ) )
51	48, 50	mpbird 247	. . . . 5 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> B ~<_ ( A \ B ) )
52		cdadom2 9009	. . . . 5 |- ( B ~<_ ( A \ B ) -> ( ( A \ B ) +c B ) ~<_ ( ( A \ B ) +c ( A \ B ) ) )
53	51, 52	syl 17	. . . 4 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( ( A \ B ) +c B ) ~<_ ( ( A \ B ) +c ( A \ B ) ) )
54		domtr 8009	. . . 4 |- ( ( A ~<_ ( ( A \ B ) +c B ) /\ ( ( A \ B ) +c B ) ~<_ ( ( A \ B ) +c ( A \ B ) ) ) -> A ~<_ ( ( A \ B ) +c ( A \ B ) ) )
55	21, 53, 54	syl2anc 693	. . 3 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> A ~<_ ( ( A \ B ) +c ( A \ B ) ) )
56		domtr 8009	. . . . . 6 |- ( ( om ~<_ A /\ A ~<_ ( ( A \ B ) +c ( A \ B ) ) ) -> om ~<_ ( ( A \ B ) +c ( A \ B ) ) )
57	29, 55, 56	syl2anc 693	. . . . 5 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> om ~<_ ( ( A \ B ) +c ( A \ B ) ) )
58		cdainf 9014	. . . . 5 |- ( om ~<_ ( A \ B ) <-> om ~<_ ( ( A \ B ) +c ( A \ B ) ) )
59	57, 58	sylibr 224	. . . 4 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> om ~<_ ( A \ B ) )
60		domrefg 7990	. . . . 5 |- ( ( A \ B ) e. dom card -> ( A \ B ) ~<_ ( A \ B ) )
61	16, 60	syl 17	. . . 4 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A \ B ) ~<_ ( A \ B ) )
62		infcdaabs 9028	. . . 4 |- ( ( ( A \ B ) e. dom card /\ om ~<_ ( A \ B ) /\ ( A \ B ) ~<_ ( A \ B ) ) -> ( ( A \ B ) +c ( A \ B ) ) ~~ ( A \ B ) )
63	16, 59, 61, 62	syl3anc 1326	. . 3 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( ( A \ B ) +c ( A \ B ) ) ~~ ( A \ B ) )
64		domentr 8015	. . 3 |- ( ( A ~<_ ( ( A \ B ) +c ( A \ B ) ) /\ ( ( A \ B ) +c ( A \ B ) ) ~~ ( A \ B ) ) -> A ~<_ ( A \ B ) )
65	55, 63, 64	syl2anc 693	. 2 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> A ~<_ ( A \ B ) )
66		sbth 8080	. 2 |- ( ( ( A \ B ) ~<_ A /\ A ~<_ ( A \ B ) ) -> ( A \ B ) ~~ A )
67	4, 65, 66	syl2anc 693	1 |- ( ( A e. dom card /\ om ~<_ A /\ B ~< A ) -> ( A \ B ) ~~ A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ w3a 1037   e. wcel 1990   \ cdif 3571   u. cun 3572   C_ wss 3574   class class class wbr 4653   dom cdm 5114  (class class class)co 6650   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761   +c ccda 8989

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

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-reu 2919  df-rmo 2920  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-int 4476  df-iun 4522  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-se 5074  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-pred 5680  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990

This theorem is referenced by:  infdif2  9032  alephsuc3  9402  aleph1irr  14975