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Mirrors > Home > ILE Home > Th. List > smoiso | Unicode version |
Description: If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.) |
Ref | Expression |
---|---|
smoiso |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isof1o 5467 | . . . 4 | |
2 | f1of 5146 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | ffdm 5081 | . . . . . 6 | |
5 | 4 | simpld 110 | . . . . 5 |
6 | fss 5074 | . . . . 5 | |
7 | 5, 6 | sylan 277 | . . . 4 |
8 | 7 | 3adant2 957 | . . 3 |
9 | 3, 8 | syl3an1 1202 | . 2 |
10 | fdm 5070 | . . . . . 6 | |
11 | 10 | eqcomd 2086 | . . . . 5 |
12 | ordeq 4127 | . . . . 5 | |
13 | 1, 2, 11, 12 | 4syl 18 | . . . 4 |
14 | 13 | biimpa 290 | . . 3 |
15 | 14 | 3adant3 958 | . 2 |
16 | 10 | eleq2d 2148 | . . . . . . 7 |
17 | 10 | eleq2d 2148 | . . . . . . 7 |
18 | 16, 17 | anbi12d 456 | . . . . . 6 |
19 | 1, 2, 18 | 3syl 17 | . . . . 5 |
20 | epel 4047 | . . . . . . . . 9 | |
21 | isorel 5468 | . . . . . . . . 9 | |
22 | 20, 21 | syl5bbr 192 | . . . . . . . 8 |
23 | ffn 5066 | . . . . . . . . . . 11 | |
24 | 3, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 270 | . . . . . . . . 9 |
26 | simprr 498 | . . . . . . . . 9 | |
27 | funfvex 5212 | . . . . . . . . . . 11 | |
28 | 27 | funfni 5019 | . . . . . . . . . 10 |
29 | epelg 4045 | . . . . . . . . . 10 | |
30 | 28, 29 | syl 14 | . . . . . . . . 9 |
31 | 25, 26, 30 | syl2anc 403 | . . . . . . . 8 |
32 | 22, 31 | bitrd 186 | . . . . . . 7 |
33 | 32 | biimpd 142 | . . . . . 6 |
34 | 33 | ex 113 | . . . . 5 |
35 | 19, 34 | sylbid 148 | . . . 4 |
36 | 35 | ralrimivv 2442 | . . 3 |
37 | 36 | 3ad2ant1 959 | . 2 |
38 | df-smo 5924 | . 2 | |
39 | 9, 15, 37, 38 | syl3anbrc 1122 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 wral 2348 cvv 2601 wss 2973 class class class wbr 3785 cep 4042 word 4117 con0 4118 cdm 4363 wfn 4917 wf 4918 wf1o 4921 cfv 4922 wiso 4923 wsmo 5923 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-tr 3876 df-eprel 4044 df-id 4048 df-iord 4121 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-f1o 4929 df-fv 4930 df-isom 4931 df-smo 5924 |
This theorem is referenced by: (None) |
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