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Mirrors > Home > ILE Home > Th. List > dfsmo2 | Unicode version |
Description: Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.) |
Ref | Expression |
---|---|
dfsmo2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-smo 5901 | . 2 | |
2 | ralcom 2473 | . . . . . 6 | |
3 | impexp 250 | . . . . . . . . 9 | |
4 | simpr 103 | . . . . . . . . . . 11 | |
5 | ordtr1 4125 | . . . . . . . . . . . . . . 15 | |
6 | 5 | 3impib 1102 | . . . . . . . . . . . . . 14 |
7 | 6 | 3com23 1110 | . . . . . . . . . . . . 13 |
8 | simp3 906 | . . . . . . . . . . . . 13 | |
9 | 7, 8 | jca 290 | . . . . . . . . . . . 12 |
10 | 9 | 3expia 1106 | . . . . . . . . . . 11 |
11 | 4, 10 | impbid2 131 | . . . . . . . . . 10 |
12 | 11 | imbi1d 220 | . . . . . . . . 9 |
13 | 3, 12 | syl5bbr 183 | . . . . . . . 8 |
14 | 13 | ralbidv2 2328 | . . . . . . 7 |
15 | 14 | ralbidva 2322 | . . . . . 6 |
16 | 2, 15 | syl5bb 181 | . . . . 5 |
17 | 16 | pm5.32i 427 | . . . 4 |
18 | 17 | anbi2i 430 | . . 3 |
19 | 3anass 889 | . . 3 | |
20 | 3anass 889 | . . 3 | |
21 | 18, 19, 20 | 3bitr4i 201 | . 2 |
22 | 1, 21 | bitri 173 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wcel 1393 wral 2306 word 4099 con0 4100 cdm 4345 wf 4898 cfv 4902 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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-tr 3855 df-iord 4103 df-smo 5901 |
This theorem is referenced by: issmo2 5904 smores2 5909 smofvon2dm 5911 |
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