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Mirrors > Home > MPE Home > Th. List > oieu | Structured version Visualization version Unicode version |
Description: Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
oicl.1 | OrdIso |
Ref | Expression |
---|---|
oieu | Se |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 796 | . . . . . 6 Se | |
2 | oicl.1 | . . . . . . . . 9 OrdIso | |
3 | 2 | ordtype 8437 | . . . . . . . 8 Se |
4 | 3 | adantr 481 | . . . . . . 7 Se |
5 | isocnv 6580 | . . . . . . 7 | |
6 | 4, 5 | syl 17 | . . . . . 6 Se |
7 | isotr 6586 | . . . . . 6 | |
8 | 1, 6, 7 | syl2anc 693 | . . . . 5 Se |
9 | simprl 794 | . . . . 5 Se | |
10 | 2 | oicl 8434 | . . . . . 6 |
11 | 10 | a1i 11 | . . . . 5 Se |
12 | ordiso2 8420 | . . . . 5 | |
13 | 8, 9, 11, 12 | syl3anc 1326 | . . . 4 Se |
14 | ordwe 5736 | . . . . . 6 | |
15 | 14 | ad2antrl 764 | . . . . 5 Se |
16 | epse 5097 | . . . . . 6 Se | |
17 | 16 | a1i 11 | . . . . 5 Se Se |
18 | isoeq4 6570 | . . . . . . 7 | |
19 | 13, 18 | syl 17 | . . . . . 6 Se |
20 | 4, 19 | mpbird 247 | . . . . 5 Se |
21 | weisoeq 6605 | . . . . 5 Se | |
22 | 15, 17, 1, 20, 21 | syl22anc 1327 | . . . 4 Se |
23 | 13, 22 | jca 554 | . . 3 Se |
24 | 23 | ex 450 | . 2 Se |
25 | 3, 10 | jctil 560 | . . 3 Se |
26 | ordeq 5730 | . . . . 5 | |
27 | 26 | adantr 481 | . . . 4 |
28 | isoeq4 6570 | . . . . 5 | |
29 | isoeq1 6567 | . . . . 5 | |
30 | 28, 29 | sylan9bb 736 | . . . 4 |
31 | 27, 30 | anbi12d 747 | . . 3 |
32 | 25, 31 | syl5ibrcom 237 | . 2 Se |
33 | 24, 32 | impbid 202 | 1 Se |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 cep 5028 Se wse 5071 wwe 5072 ccnv 5113 cdm 5114 ccom 5118 word 5722 wiso 5889 OrdIsocoi 8414 |
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-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-wrecs 7407 df-recs 7468 df-oi 8415 |
This theorem is referenced by: hartogslem1 8447 cantnfp1lem3 8577 oemapwe 8591 cantnffval2 8592 om2uzoi 12754 |
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