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Mirrors > Home > MPE Home > Th. List > sucdom2 | Structured version Visualization version Unicode version |
Description: Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
sucdom2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomdom 7983 | . . 3 | |
2 | brdomi 7966 | . . 3 | |
3 | 1, 2 | syl 17 | . 2 |
4 | relsdom 7962 | . . . . . . 7 | |
5 | 4 | brrelexi 5158 | . . . . . 6 |
6 | 5 | adantr 481 | . . . . 5 |
7 | vex 3203 | . . . . . . 7 | |
8 | 7 | rnex 7100 | . . . . . 6 |
9 | 8 | a1i 11 | . . . . 5 |
10 | f1f1orn 6148 | . . . . . . 7 | |
11 | 10 | adantl 482 | . . . . . 6 |
12 | f1of1 6136 | . . . . . 6 | |
13 | 11, 12 | syl 17 | . . . . 5 |
14 | f1dom2g 7973 | . . . . 5 | |
15 | 6, 9, 13, 14 | syl3anc 1326 | . . . 4 |
16 | sdomnen 7984 | . . . . . . . 8 | |
17 | 16 | adantr 481 | . . . . . . 7 |
18 | ssdif0 3942 | . . . . . . . 8 | |
19 | simplr 792 | . . . . . . . . . . 11 | |
20 | f1f 6101 | . . . . . . . . . . . . . . 15 | |
21 | df-f 5892 | . . . . . . . . . . . . . . 15 | |
22 | 20, 21 | sylib 208 | . . . . . . . . . . . . . 14 |
23 | 22 | simprd 479 | . . . . . . . . . . . . 13 |
24 | 19, 23 | syl 17 | . . . . . . . . . . . 12 |
25 | simpr 477 | . . . . . . . . . . . 12 | |
26 | 24, 25 | eqssd 3620 | . . . . . . . . . . 11 |
27 | dff1o5 6146 | . . . . . . . . . . 11 | |
28 | 19, 26, 27 | sylanbrc 698 | . . . . . . . . . 10 |
29 | f1oen3g 7971 | . . . . . . . . . 10 | |
30 | 7, 28, 29 | sylancr 695 | . . . . . . . . 9 |
31 | 30 | ex 450 | . . . . . . . 8 |
32 | 18, 31 | syl5bir 233 | . . . . . . 7 |
33 | 17, 32 | mtod 189 | . . . . . 6 |
34 | neq0 3930 | . . . . . 6 | |
35 | 33, 34 | sylib 208 | . . . . 5 |
36 | snssi 4339 | . . . . . . 7 | |
37 | vex 3203 | . . . . . . . . 9 | |
38 | en2sn 8037 | . . . . . . . . 9 | |
39 | 6, 37, 38 | sylancl 694 | . . . . . . . 8 |
40 | 4 | brrelex2i 5159 | . . . . . . . . . 10 |
41 | 40 | adantr 481 | . . . . . . . . 9 |
42 | difexg 4808 | . . . . . . . . 9 | |
43 | ssdomg 8001 | . . . . . . . . 9 | |
44 | 41, 42, 43 | 3syl 18 | . . . . . . . 8 |
45 | endomtr 8014 | . . . . . . . 8 | |
46 | 39, 44, 45 | syl6an 568 | . . . . . . 7 |
47 | 36, 46 | syl5 34 | . . . . . 6 |
48 | 47 | exlimdv 1861 | . . . . 5 |
49 | 35, 48 | mpd 15 | . . . 4 |
50 | disjdif 4040 | . . . . 5 | |
51 | 50 | a1i 11 | . . . 4 |
52 | undom 8048 | . . . 4 | |
53 | 15, 49, 51, 52 | syl21anc 1325 | . . 3 |
54 | df-suc 5729 | . . . 4 | |
55 | 54 | a1i 11 | . . 3 |
56 | undif2 4044 | . . . 4 | |
57 | 23 | adantl 482 | . . . . 5 |
58 | ssequn1 3783 | . . . . 5 | |
59 | 57, 58 | sylib 208 | . . . 4 |
60 | 56, 59 | syl5req 2669 | . . 3 |
61 | 53, 55, 60 | 3brtr4d 4685 | . 2 |
62 | 3, 61 | exlimddv 1863 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wex 1704 wcel 1990 cvv 3200 cdif 3571 cun 3572 cin 3573 wss 3574 c0 3915 csn 4177 class class class wbr 4653 crn 5115 csuc 5725 wfn 5883 wf 5884 wf1 5885 wf1o 5887 cen 7952 cdom 7953 csdm 7954 |
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-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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 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-suc 5729 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 |
This theorem is referenced by: sucdom 8157 card2inf 8460 |
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