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Mirrors > Home > ILE Home > Th. List > issmo | Unicode version |
Description: Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Ref | Expression |
---|---|
issmo.1 | |
issmo.2 | |
issmo.3 | |
issmo.4 |
Ref | Expression |
---|---|
issmo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmo.1 | . . 3 | |
2 | issmo.4 | . . . 4 | |
3 | 2 | feq2i 5236 | . . 3 |
4 | 1, 3 | mpbir 145 | . 2 |
5 | issmo.2 | . . 3 | |
6 | ordeq 4264 | . . . 4 | |
7 | 2, 6 | ax-mp 5 | . . 3 |
8 | 5, 7 | mpbir 145 | . 2 |
9 | 2 | eleq2i 2184 | . . . 4 |
10 | 2 | eleq2i 2184 | . . . 4 |
11 | issmo.3 | . . . 4 | |
12 | 9, 10, 11 | syl2anb 289 | . . 3 |
13 | 12 | rgen2a 2463 | . 2 |
14 | df-smo 6151 | . 2 | |
15 | 4, 8, 13, 14 | mpbir3an 1148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 word 4254 con0 4255 cdm 4509 wf 5089 cfv 5093 wsmo 6150 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 df-fn 5096 df-f 5097 df-smo 6151 |
This theorem is referenced by: iordsmo 6162 |
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