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Mirrors > Home > ILE Home > Th. List > ordpwsucexmid | Unicode version |
Description: The subset in ordpwsucss 4452 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 4058 | . . . . 5 | |
2 | 0elon 4284 | . . . . 5 | |
3 | elin 3229 | . . . . 5 | |
4 | 1, 2, 3 | mpbir2an 911 | . . . 4 |
5 | ordtriexmidlem 4405 | . . . . 5 | |
6 | suceq 4294 | . . . . . . 7 | |
7 | pweq 3483 | . . . . . . . 8 | |
8 | 7 | ineq1d 3246 | . . . . . . 7 |
9 | 6, 8 | eqeq12d 2132 | . . . . . 6 |
10 | ordpwsucexmid.1 | . . . . . 6 | |
11 | 9, 10 | vtoclri 2735 | . . . . 5 |
12 | 5, 11 | ax-mp 5 | . . . 4 |
13 | 4, 12 | eleqtrri 2193 | . . 3 |
14 | elsuci 4295 | . . 3 | |
15 | 13, 14 | ax-mp 5 | . 2 |
16 | 0ex 4025 | . . . . . 6 | |
17 | 16 | snid 3526 | . . . . 5 |
18 | biidd 171 | . . . . . 6 | |
19 | 18 | elrab3 2814 | . . . . 5 |
20 | 17, 19 | ax-mp 5 | . . . 4 |
21 | 20 | biimpi 119 | . . 3 |
22 | ordtriexmidlem2 4406 | . . . 4 | |
23 | 22 | eqcoms 2120 | . . 3 |
24 | 21, 23 | orim12i 733 | . 2 |
25 | 15, 24 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 104 wo 682 wceq 1316 wcel 1465 wral 2393 crab 2397 cin 3040 c0 3333 cpw 3480 csn 3497 con0 4255 csuc 4257 |
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-in1 588 ax-in2 589 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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 |
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-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 |
This theorem is referenced by: (None) |
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