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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 5701 | . . . 4 | |
2 | f1of 5360 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | ffdm 5288 | . . . . . 6 | |
5 | 4 | simpld 111 | . . . . 5 |
6 | fss 5279 | . . . . 5 | |
7 | 5, 6 | sylan 281 | . . . 4 |
8 | 7 | 3adant2 1000 | . . 3 |
9 | 3, 8 | syl3an1 1249 | . 2 |
10 | fdm 5273 | . . . . . 6 | |
11 | 10 | eqcomd 2143 | . . . . 5 |
12 | ordeq 4289 | . . . . 5 | |
13 | 1, 2, 11, 12 | 4syl 18 | . . . 4 |
14 | 13 | biimpa 294 | . . 3 |
15 | 14 | 3adant3 1001 | . 2 |
16 | 10 | eleq2d 2207 | . . . . . . 7 |
17 | 10 | eleq2d 2207 | . . . . . . 7 |
18 | 16, 17 | anbi12d 464 | . . . . . 6 |
19 | 1, 2, 18 | 3syl 17 | . . . . 5 |
20 | epel 4209 | . . . . . . . . 9 | |
21 | isorel 5702 | . . . . . . . . 9 | |
22 | 20, 21 | syl5bbr 193 | . . . . . . . 8 |
23 | ffn 5267 | . . . . . . . . . . 11 | |
24 | 3, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 274 | . . . . . . . . 9 |
26 | simprr 521 | . . . . . . . . 9 | |
27 | funfvex 5431 | . . . . . . . . . . 11 | |
28 | 27 | funfni 5218 | . . . . . . . . . 10 |
29 | epelg 4207 | . . . . . . . . . 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 2511 | . . 3 |
37 | 36 | 3ad2ant1 1002 | . 2 |
38 | df-smo 6176 | . 2 | |
39 | 9, 15, 37, 38 | syl3anbrc 1165 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2414 cvv 2681 wss 3066 class class class wbr 3924 cep 4204 word 4279 con0 4280 cdm 4534 wfn 5113 wf 5114 wf1o 5117 cfv 5118 wiso 5119 wsmo 6175 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-tr 4022 df-eprel 4206 df-id 4210 df-iord 4283 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-f1o 5125 df-fv 5126 df-isom 5127 df-smo 6176 |
This theorem is referenced by: (None) |
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