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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 5716 | . . . 4 | |
2 | f1of 5375 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | ffdm 5301 | . . . . . 6 | |
5 | 4 | simpld 111 | . . . . 5 |
6 | fss 5292 | . . . . 5 | |
7 | 5, 6 | sylan 281 | . . . 4 |
8 | 7 | 3adant2 1001 | . . 3 |
9 | 3, 8 | syl3an1 1250 | . 2 |
10 | fdm 5286 | . . . . . 6 | |
11 | 10 | eqcomd 2146 | . . . . 5 |
12 | ordeq 4302 | . . . . 5 | |
13 | 1, 2, 11, 12 | 4syl 18 | . . . 4 |
14 | 13 | biimpa 294 | . . 3 |
15 | 14 | 3adant3 1002 | . 2 |
16 | 10 | eleq2d 2210 | . . . . . . 7 |
17 | 10 | eleq2d 2210 | . . . . . . 7 |
18 | 16, 17 | anbi12d 465 | . . . . . 6 |
19 | 1, 2, 18 | 3syl 17 | . . . . 5 |
20 | epel 4222 | . . . . . . . . 9 | |
21 | isorel 5717 | . . . . . . . . 9 | |
22 | 20, 21 | bitr3id 193 | . . . . . . . 8 |
23 | ffn 5280 | . . . . . . . . . . 11 | |
24 | 3, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 274 | . . . . . . . . 9 |
26 | simprr 522 | . . . . . . . . 9 | |
27 | funfvex 5446 | . . . . . . . . . . 11 | |
28 | 27 | funfni 5231 | . . . . . . . . . 10 |
29 | epelg 4220 | . . . . . . . . . 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 2516 | . . 3 |
37 | 36 | 3ad2ant1 1003 | . 2 |
38 | df-smo 6191 | . 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 1332 wcel 1481 wral 2417 cvv 2689 wss 3076 class class class wbr 3937 cep 4217 word 4292 con0 4293 cdm 4547 wfn 5126 wf 5127 wf1o 5130 cfv 5131 wiso 5132 wsmo 6190 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-tr 4035 df-eprel 4219 df-id 4223 df-iord 4296 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-f1o 5138 df-fv 5139 df-isom 5140 df-smo 6191 |
This theorem is referenced by: (None) |
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