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Mirrors > Home > MPE Home > Th. List > dfsmo2 | Structured version Visualization version 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 7443 | . 2 | |
2 | ralcom 3098 | . . . . . 6 | |
3 | impexp 462 | . . . . . . . . 9 | |
4 | simpr 477 | . . . . . . . . . . 11 | |
5 | ordtr1 5767 | . . . . . . . . . . . . . . 15 | |
6 | 5 | 3impib 1262 | . . . . . . . . . . . . . 14 |
7 | 6 | 3com23 1271 | . . . . . . . . . . . . 13 |
8 | simp3 1063 | . . . . . . . . . . . . 13 | |
9 | 7, 8 | jca 554 | . . . . . . . . . . . 12 |
10 | 9 | 3expia 1267 | . . . . . . . . . . 11 |
11 | 4, 10 | impbid2 216 | . . . . . . . . . 10 |
12 | 11 | imbi1d 331 | . . . . . . . . 9 |
13 | 3, 12 | syl5bbr 274 | . . . . . . . 8 |
14 | 13 | ralbidv2 2984 | . . . . . . 7 |
15 | 14 | ralbidva 2985 | . . . . . 6 |
16 | 2, 15 | syl5bb 272 | . . . . 5 |
17 | 16 | pm5.32i 669 | . . . 4 |
18 | 17 | anbi2i 730 | . . 3 |
19 | 3anass 1042 | . . 3 | |
20 | 3anass 1042 | . . 3 | |
21 | 18, 19, 20 | 3bitr4i 292 | . 2 |
22 | 1, 21 | bitri 264 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wcel 1990 wral 2912 cdm 5114 word 5722 con0 5723 wf 5884 cfv 5888 wsmo 7442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 df-in 3581 df-ss 3588 df-uni 4437 df-tr 4753 df-ord 5726 df-smo 7443 |
This theorem is referenced by: issmo2 7446 smores2 7451 smofvon2 7453 |
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