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Mirrors > Home > MPE Home > Th. List > finnisoeu | Structured version Visualization version Unicode version |
Description: A finite totally ordered set has a unique order isomorphism to a finite ordinal. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
finnisoeu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . 5 OrdIso OrdIso | |
2 | 1 | oiexg 8440 | . . . 4 OrdIso |
3 | 2 | adantl 482 | . . 3 OrdIso |
4 | simpr 477 | . . . . 5 | |
5 | wofi 8209 | . . . . 5 | |
6 | 1 | oiiso 8442 | . . . . 5 OrdIso OrdIso |
7 | 4, 5, 6 | syl2anc 693 | . . . 4 OrdIso OrdIso |
8 | 1 | oien 8443 | . . . . . . . 8 OrdIso |
9 | 4, 5, 8 | syl2anc 693 | . . . . . . 7 OrdIso |
10 | ficardid 8788 | . . . . . . . . 9 | |
11 | 10 | adantl 482 | . . . . . . . 8 |
12 | 11 | ensymd 8007 | . . . . . . 7 |
13 | entr 8008 | . . . . . . 7 OrdIso OrdIso | |
14 | 9, 12, 13 | syl2anc 693 | . . . . . 6 OrdIso |
15 | 1 | oion 8441 | . . . . . . . 8 OrdIso |
16 | 15 | adantl 482 | . . . . . . 7 OrdIso |
17 | ficardom 8787 | . . . . . . . 8 | |
18 | 17 | adantl 482 | . . . . . . 7 |
19 | onomeneq 8150 | . . . . . . 7 OrdIso OrdIso OrdIso | |
20 | 16, 18, 19 | syl2anc 693 | . . . . . 6 OrdIso OrdIso |
21 | 14, 20 | mpbid 222 | . . . . 5 OrdIso |
22 | isoeq4 6570 | . . . . 5 OrdIso OrdIso OrdIso OrdIso | |
23 | 21, 22 | syl 17 | . . . 4 OrdIso OrdIso OrdIso |
24 | 7, 23 | mpbid 222 | . . 3 OrdIso |
25 | isoeq1 6567 | . . . 4 OrdIso OrdIso | |
26 | 25 | spcegv 3294 | . . 3 OrdIso OrdIso |
27 | 3, 24, 26 | sylc 65 | . 2 |
28 | wemoiso2 7154 | . . 3 | |
29 | 5, 28 | syl 17 | . 2 |
30 | eu5 2496 | . 2 | |
31 | 27, 29, 30 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 weu 2470 wmo 2471 cvv 3200 class class class wbr 4653 cep 5028 wor 5034 wwe 5072 cdm 5114 con0 5723 cfv 5888 wiso 5889 com 7065 cen 7952 cfn 7955 OrdIsocoi 8414 ccrd 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-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-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-om 7066 df-wrecs 7407 df-recs 7468 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-card 8765 |
This theorem is referenced by: iunfictbso 8937 |
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