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Mirrors > Home > MPE Home > Th. List > isfin4-3 | Structured version Visualization version Unicode version |
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9119 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
isfin4-3 | FinIV |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 7567 | . . . 4 | |
2 | cdadom3 9010 | . . . 4 FinIV | |
3 | 1, 2 | mpan2 707 | . . 3 FinIV |
4 | ssun1 3776 | . . . . . . . 8 | |
5 | relen 7960 | . . . . . . . . . 10 | |
6 | 5 | brrelexi 5158 | . . . . . . . . 9 |
7 | cdaval 8992 | . . . . . . . . 9 | |
8 | 6, 1, 7 | sylancl 694 | . . . . . . . 8 |
9 | 4, 8 | syl5sseqr 3654 | . . . . . . 7 |
10 | 0lt1o 7584 | . . . . . . . . . 10 | |
11 | 1 | elexi 3213 | . . . . . . . . . . 11 |
12 | 11 | snid 4208 | . . . . . . . . . 10 |
13 | opelxpi 5148 | . . . . . . . . . 10 | |
14 | 10, 12, 13 | mp2an 708 | . . . . . . . . 9 |
15 | elun2 3781 | . . . . . . . . 9 | |
16 | 14, 15 | mp1i 13 | . . . . . . . 8 |
17 | 16, 8 | eleqtrrd 2704 | . . . . . . 7 |
18 | 1n0 7575 | . . . . . . . 8 | |
19 | opelxp2 5151 | . . . . . . . . . 10 | |
20 | elsni 4194 | . . . . . . . . . 10 | |
21 | 19, 20 | syl 17 | . . . . . . . . 9 |
22 | 21 | necon3ai 2819 | . . . . . . . 8 |
23 | 18, 22 | mp1i 13 | . . . . . . 7 |
24 | 9, 17, 23 | ssnelpssd 3719 | . . . . . 6 |
25 | 0ex 4790 | . . . . . . . 8 | |
26 | xpsneng 8045 | . . . . . . . 8 | |
27 | 6, 25, 26 | sylancl 694 | . . . . . . 7 |
28 | entr 8008 | . . . . . . 7 | |
29 | 27, 28 | mpancom 703 | . . . . . 6 |
30 | fin4i 9120 | . . . . . 6 FinIV | |
31 | 24, 29, 30 | syl2anc 693 | . . . . 5 FinIV |
32 | fin4en1 9131 | . . . . 5 FinIV FinIV | |
33 | 31, 32 | mtod 189 | . . . 4 FinIV |
34 | 33 | con2i 134 | . . 3 FinIV |
35 | brsdom 7978 | . . 3 | |
36 | 3, 34, 35 | sylanbrc 698 | . 2 FinIV |
37 | sdomnen 7984 | . . . 4 | |
38 | infcda1 9015 | . . . . 5 | |
39 | 38 | ensymd 8007 | . . . 4 |
40 | 37, 39 | nsyl 135 | . . 3 |
41 | relsdom 7962 | . . . . 5 | |
42 | 41 | brrelexi 5158 | . . . 4 |
43 | isfin4-2 9136 | . . . 4 FinIV | |
44 | 42, 43 | syl 17 | . . 3 FinIV |
45 | 40, 44 | mpbird 247 | . 2 FinIV |
46 | 36, 45 | impbii 199 | 1 FinIV |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 196 wceq 1483 wcel 1990 wne 2794 cvv 3200 cun 3572 wpss 3575 c0 3915 csn 4177 cop 4183 class class class wbr 4653 cxp 5112 con0 5723 (class class class)co 6650 com 7065 c1o 7553 cen 7952 cdom 7953 csdm 7954 ccda 8989 FinIVcfin4 9102 |
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 |
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-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-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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-cda 8990 df-fin4 9109 |
This theorem is referenced by: fin45 9214 finngch 9477 gchinf 9479 |
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