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Mirrors > Home > ILE Home > Th. List > ordpwsucexmid | Unicode version |
Description: The subset in ordpwsucss 4551 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 4150 | . . . . 5 | |
2 | 0elon 4377 | . . . . 5 | |
3 | elin 3310 | . . . . 5 | |
4 | 1, 2, 3 | mpbir2an 937 | . . . 4 |
5 | ordtriexmidlem 4503 | . . . . 5 | |
6 | suceq 4387 | . . . . . . 7 | |
7 | pweq 3569 | . . . . . . . 8 | |
8 | 7 | ineq1d 3327 | . . . . . . 7 |
9 | 6, 8 | eqeq12d 2185 | . . . . . 6 |
10 | ordpwsucexmid.1 | . . . . . 6 | |
11 | 9, 10 | vtoclri 2805 | . . . . 5 |
12 | 5, 11 | ax-mp 5 | . . . 4 |
13 | 4, 12 | eleqtrri 2246 | . . 3 |
14 | elsuci 4388 | . . 3 | |
15 | 13, 14 | ax-mp 5 | . 2 |
16 | 0ex 4116 | . . . . . 6 | |
17 | 16 | snid 3614 | . . . . 5 |
18 | biidd 171 | . . . . . 6 | |
19 | 18 | elrab3 2887 | . . . . 5 |
20 | 17, 19 | ax-mp 5 | . . . 4 |
21 | 20 | biimpi 119 | . . 3 |
22 | ordtriexmidlem2 4504 | . . . 4 | |
23 | 22 | eqcoms 2173 | . . 3 |
24 | 21, 23 | orim12i 754 | . 2 |
25 | 15, 24 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 104 wo 703 wceq 1348 wcel 2141 wral 2448 crab 2452 cin 3120 c0 3414 cpw 3566 csn 3583 con0 4348 csuc 4350 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 |
This theorem is referenced by: (None) |
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