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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 5676 | . . . 4 | |
2 | f1of 5335 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | ffdm 5263 | . . . . . 6 | |
5 | 4 | simpld 111 | . . . . 5 |
6 | fss 5254 | . . . . 5 | |
7 | 5, 6 | sylan 281 | . . . 4 |
8 | 7 | 3adant2 985 | . . 3 |
9 | 3, 8 | syl3an1 1234 | . 2 |
10 | fdm 5248 | . . . . . 6 | |
11 | 10 | eqcomd 2123 | . . . . 5 |
12 | ordeq 4264 | . . . . 5 | |
13 | 1, 2, 11, 12 | 4syl 18 | . . . 4 |
14 | 13 | biimpa 294 | . . 3 |
15 | 14 | 3adant3 986 | . 2 |
16 | 10 | eleq2d 2187 | . . . . . . 7 |
17 | 10 | eleq2d 2187 | . . . . . . 7 |
18 | 16, 17 | anbi12d 464 | . . . . . 6 |
19 | 1, 2, 18 | 3syl 17 | . . . . 5 |
20 | epel 4184 | . . . . . . . . 9 | |
21 | isorel 5677 | . . . . . . . . 9 | |
22 | 20, 21 | syl5bbr 193 | . . . . . . . 8 |
23 | ffn 5242 | . . . . . . . . . . 11 | |
24 | 3, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 274 | . . . . . . . . 9 |
26 | simprr 506 | . . . . . . . . 9 | |
27 | funfvex 5406 | . . . . . . . . . . 11 | |
28 | 27 | funfni 5193 | . . . . . . . . . 10 |
29 | epelg 4182 | . . . . . . . . . 10 | |
30 | 28, 29 | syl 14 | . . . . . . . . 9 |
31 | 25, 26, 30 | syl2anc 408 | . . . . . . . 8 |
32 | 22, 31 | bitrd 187 | . . . . . . 7 |
33 | 32 | biimpd 143 | . . . . . 6 |
34 | 33 | ex 114 | . . . . 5 |
35 | 19, 34 | sylbid 149 | . . . 4 |
36 | 35 | ralrimivv 2490 | . . 3 |
37 | 36 | 3ad2ant1 987 | . 2 |
38 | df-smo 6151 | . 2 | |
39 | 9, 15, 37, 38 | syl3anbrc 1150 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wral 2393 cvv 2660 wss 3041 class class class wbr 3899 cep 4179 word 4254 con0 4255 cdm 4509 wfn 5088 wf 5089 wf1o 5092 cfv 5093 wiso 5094 wsmo 6150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-tr 3997 df-eprel 4181 df-id 4185 df-iord 4258 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-f1o 5100 df-fv 5101 df-isom 5102 df-smo 6151 |
This theorem is referenced by: (None) |
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