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Mirrors > Home > ILE Home > Th. List > smores3 | Unicode version |
Description: A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
smores3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 4886 | . . . . . 6 | |
2 | incom 3299 | . . . . . 6 | |
3 | 1, 2 | eqtri 2178 | . . . . 5 |
4 | 3 | eleq2i 2224 | . . . 4 |
5 | smores 6236 | . . . 4 | |
6 | 4, 5 | sylan2br 286 | . . 3 |
7 | 6 | 3adant3 1002 | . 2 |
8 | inss2 3328 | . . . . . 6 | |
9 | 8 | sseli 3124 | . . . . 5 |
10 | ordelss 4339 | . . . . . 6 | |
11 | 10 | ancoms 266 | . . . . 5 |
12 | 9, 11 | sylan 281 | . . . 4 |
13 | 12 | 3adant1 1000 | . . 3 |
14 | resabs1 4894 | . . 3 | |
15 | smoeq 6234 | . . 3 | |
16 | 13, 14, 15 | 3syl 17 | . 2 |
17 | 7, 16 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 963 wceq 1335 wcel 2128 cin 3101 wss 3102 word 4322 cdm 4585 cres 4587 wsmo 6229 |
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-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 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-iord 4326 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-fv 5177 df-smo 6230 |
This theorem is referenced by: (None) |
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