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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 5757 | . . . 4 | |
2 | f1of 5414 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | ffdm 5340 | . . . . . 6 | |
5 | 4 | simpld 111 | . . . . 5 |
6 | fss 5331 | . . . . 5 | |
7 | 5, 6 | sylan 281 | . . . 4 |
8 | 7 | 3adant2 1001 | . . 3 |
9 | 3, 8 | syl3an1 1253 | . 2 |
10 | fdm 5325 | . . . . . 6 | |
11 | 10 | eqcomd 2163 | . . . . 5 |
12 | ordeq 4332 | . . . . 5 | |
13 | 1, 2, 11, 12 | 4syl 18 | . . . 4 |
14 | 13 | biimpa 294 | . . 3 |
15 | 14 | 3adant3 1002 | . 2 |
16 | 10 | eleq2d 2227 | . . . . . . 7 |
17 | 10 | eleq2d 2227 | . . . . . . 7 |
18 | 16, 17 | anbi12d 465 | . . . . . 6 |
19 | 1, 2, 18 | 3syl 17 | . . . . 5 |
20 | epel 4252 | . . . . . . . . 9 | |
21 | isorel 5758 | . . . . . . . . 9 | |
22 | 20, 21 | bitr3id 193 | . . . . . . . 8 |
23 | ffn 5319 | . . . . . . . . . . 11 | |
24 | 3, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 274 | . . . . . . . . 9 |
26 | simprr 522 | . . . . . . . . 9 | |
27 | funfvex 5485 | . . . . . . . . . . 11 | |
28 | 27 | funfni 5270 | . . . . . . . . . 10 |
29 | epelg 4250 | . . . . . . . . . 10 | |
30 | 28, 29 | syl 14 | . . . . . . . . 9 |
31 | 25, 26, 30 | syl2anc 409 | . . . . . . . 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 2538 | . . 3 |
37 | 36 | 3ad2ant1 1003 | . 2 |
38 | df-smo 6233 | . 2 | |
39 | 9, 15, 37, 38 | syl3anbrc 1166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 wral 2435 cvv 2712 wss 3102 class class class wbr 3965 cep 4247 word 4322 con0 4323 cdm 4586 wfn 5165 wf 5166 wf1o 5169 cfv 5170 wiso 5171 wsmo 6232 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-tr 4063 df-eprel 4249 df-id 4253 df-iord 4326 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-f1o 5177 df-fv 5178 df-isom 5179 df-smo 6233 |
This theorem is referenced by: (None) |
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