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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 5447 | . . . 4 | |
2 | f1of 5126 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | ffdm 5061 | . . . . . 6 | |
5 | 4 | simpld 105 | . . . . 5 |
6 | fss 5054 | . . . . 5 | |
7 | 5, 6 | sylan 267 | . . . 4 |
8 | 7 | 3adant2 923 | . . 3 |
9 | 3, 8 | syl3an1 1168 | . 2 |
10 | fdm 5050 | . . . . . 6 | |
11 | 10 | eqcomd 2045 | . . . . 5 |
12 | ordeq 4109 | . . . . 5 | |
13 | 1, 2, 11, 12 | 4syl 18 | . . . 4 |
14 | 13 | biimpa 280 | . . 3 |
15 | 14 | 3adant3 924 | . 2 |
16 | 10 | eleq2d 2107 | . . . . . . 7 |
17 | 10 | eleq2d 2107 | . . . . . . 7 |
18 | 16, 17 | anbi12d 442 | . . . . . 6 |
19 | 1, 2, 18 | 3syl 17 | . . . . 5 |
20 | epel 4029 | . . . . . . . . 9 | |
21 | isorel 5448 | . . . . . . . . 9 | |
22 | 20, 21 | syl5bbr 183 | . . . . . . . 8 |
23 | ffn 5046 | . . . . . . . . . . 11 | |
24 | 3, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 261 | . . . . . . . . 9 |
26 | simprr 484 | . . . . . . . . 9 | |
27 | funfvex 5192 | . . . . . . . . . . 11 | |
28 | 27 | funfni 4999 | . . . . . . . . . 10 |
29 | epelg 4027 | . . . . . . . . . 10 | |
30 | 28, 29 | syl 14 | . . . . . . . . 9 |
31 | 25, 26, 30 | syl2anc 391 | . . . . . . . 8 |
32 | 22, 31 | bitrd 177 | . . . . . . 7 |
33 | 32 | biimpd 132 | . . . . . 6 |
34 | 33 | ex 108 | . . . . 5 |
35 | 19, 34 | sylbid 139 | . . . 4 |
36 | 35 | ralrimivv 2400 | . . 3 |
37 | 36 | 3ad2ant1 925 | . 2 |
38 | df-smo 5901 | . 2 | |
39 | 9, 15, 37, 38 | syl3anbrc 1088 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 wral 2306 cvv 2557 wss 2917 class class class wbr 3764 cep 4024 word 4099 con0 4100 cdm 4345 wfn 4897 wf 4898 wf1o 4901 cfv 4902 wiso 4903 wsmo 5900 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-tr 3855 df-eprel 4026 df-id 4030 df-iord 4103 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-f1o 4909 df-fv 4910 df-isom 4911 df-smo 5901 |
This theorem is referenced by: (None) |
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