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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 5040 | . . 3 |
4 | 1, 3 | mpbir 134 | . 2 |
5 | issmo.2 | . . 3 | |
6 | ordeq 4109 | . . . 4 | |
7 | 2, 6 | ax-mp 7 | . . 3 |
8 | 5, 7 | mpbir 134 | . 2 |
9 | 2 | eleq2i 2104 | . . . 4 |
10 | 2 | eleq2i 2104 | . . . 4 |
11 | issmo.3 | . . . 4 | |
12 | 9, 10, 11 | syl2anb 275 | . . 3 |
13 | 12 | rgen2a 2375 | . 2 |
14 | df-smo 5901 | . 2 | |
15 | 4, 8, 13, 14 | mpbir3an 1086 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 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-rex 2312 df-in 2924 df-ss 2931 df-uni 3581 df-tr 3855 df-iord 4103 df-fn 4905 df-f 4906 df-smo 5901 |
This theorem is referenced by: iordsmo 5912 |
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