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Mirrors > Home > MPE Home > Th. List > isfin7-2 | Structured version Visualization version Unicode version |
Description: A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
isfin7-2 | FinVII |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfin7 9123 | . . . 4 FinVII FinVII | |
2 | 1 | ibi 256 | . . 3 FinVII |
3 | isnum2 8771 | . . . . 5 | |
4 | ensym 8005 | . . . . . . . . 9 | |
5 | simprl 794 | . . . . . . . . . . 11 | |
6 | enfi 8176 | . . . . . . . . . . . . . . 15 | |
7 | onfin 8151 | . . . . . . . . . . . . . . 15 | |
8 | 6, 7 | sylan9bbr 737 | . . . . . . . . . . . . . 14 |
9 | 8 | biimprd 238 | . . . . . . . . . . . . 13 |
10 | 9 | con3d 148 | . . . . . . . . . . . 12 |
11 | 10 | impcom 446 | . . . . . . . . . . 11 |
12 | 5, 11 | eldifd 3585 | . . . . . . . . . 10 |
13 | simprr 796 | . . . . . . . . . 10 | |
14 | 12, 13 | jca 554 | . . . . . . . . 9 |
15 | 4, 14 | sylanr2 685 | . . . . . . . 8 |
16 | 15 | ex 450 | . . . . . . 7 |
17 | 16 | reximdv2 3014 | . . . . . 6 |
18 | 17 | com12 32 | . . . . 5 |
19 | 3, 18 | sylbi 207 | . . . 4 |
20 | 19 | con1d 139 | . . 3 |
21 | 2, 20 | syl5com 31 | . 2 FinVII |
22 | eldifi 3732 | . . . . . . 7 | |
23 | ensym 8005 | . . . . . . 7 | |
24 | isnumi 8772 | . . . . . . 7 | |
25 | 22, 23, 24 | syl2an 494 | . . . . . 6 |
26 | 25 | rexlimiva 3028 | . . . . 5 |
27 | 26 | con3i 150 | . . . 4 |
28 | isfin7 9123 | . . . 4 FinVII | |
29 | 27, 28 | syl5ibr 236 | . . 3 FinVII |
30 | fin17 9216 | . . . 4 FinVII | |
31 | 30 | a1i 11 | . . 3 FinVII |
32 | 29, 31 | jad 174 | . 2 FinVII |
33 | 21, 32 | impbid2 216 | 1 FinVII |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wcel 1990 wrex 2913 cdif 3571 class class class wbr 4653 cdm 5114 con0 5723 com 7065 cen 7952 cfn 7955 ccrd 8761 FinVIIcfin7 9106 |
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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-fin7 9113 |
This theorem is referenced by: fin71num 9219 dffin7-2 9220 |
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