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Mirrors > Home > ILE Home > Th. List > ordpwsucexmid | Unicode version |
Description: The subset in ordpwsucss 4310 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
Ref | Expression |
---|---|
ordpwsucexmid.1 |
Ref | Expression |
---|---|
ordpwsucexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 3938 | . . . . 5 | |
2 | 0elon 4147 | . . . . 5 | |
3 | elin 3155 | . . . . 5 | |
4 | 1, 2, 3 | mpbir2an 883 | . . . 4 |
5 | ordtriexmidlem 4263 | . . . . 5 | |
6 | suceq 4157 | . . . . . . 7 | |
7 | pweq 3385 | . . . . . . . 8 | |
8 | 7 | ineq1d 3166 | . . . . . . 7 |
9 | 6, 8 | eqeq12d 2095 | . . . . . 6 |
10 | ordpwsucexmid.1 | . . . . . 6 | |
11 | 9, 10 | vtoclri 2673 | . . . . 5 |
12 | 5, 11 | ax-mp 7 | . . . 4 |
13 | 4, 12 | eleqtrri 2154 | . . 3 |
14 | elsuci 4158 | . . 3 | |
15 | 13, 14 | ax-mp 7 | . 2 |
16 | 0ex 3905 | . . . . . 6 | |
17 | 16 | snid 3425 | . . . . 5 |
18 | biidd 170 | . . . . . 6 | |
19 | 18 | elrab3 2750 | . . . . 5 |
20 | 17, 19 | ax-mp 7 | . . . 4 |
21 | 20 | biimpi 118 | . . 3 |
22 | ordtriexmidlem2 4264 | . . . 4 | |
23 | 22 | eqcoms 2084 | . . 3 |
24 | 21, 23 | orim12i 708 | . 2 |
25 | 15, 24 | ax-mp 7 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 103 wo 661 wceq 1284 wcel 1433 wral 2348 crab 2352 cin 2972 c0 3251 cpw 3382 csn 3398 con0 4118 csuc 4120 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 |
This theorem is referenced by: (None) |
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