Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ordpwsucexmid | Unicode version |
Description: The subset in ordpwsucss 4527 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 4126 | . . . . 5 | |
2 | 0elon 4353 | . . . . 5 | |
3 | elin 3290 | . . . . 5 | |
4 | 1, 2, 3 | mpbir2an 927 | . . . 4 |
5 | ordtriexmidlem 4479 | . . . . 5 | |
6 | suceq 4363 | . . . . . . 7 | |
7 | pweq 3546 | . . . . . . . 8 | |
8 | 7 | ineq1d 3307 | . . . . . . 7 |
9 | 6, 8 | eqeq12d 2172 | . . . . . 6 |
10 | ordpwsucexmid.1 | . . . . . 6 | |
11 | 9, 10 | vtoclri 2787 | . . . . 5 |
12 | 5, 11 | ax-mp 5 | . . . 4 |
13 | 4, 12 | eleqtrri 2233 | . . 3 |
14 | elsuci 4364 | . . 3 | |
15 | 13, 14 | ax-mp 5 | . 2 |
16 | 0ex 4092 | . . . . . 6 | |
17 | 16 | snid 3591 | . . . . 5 |
18 | biidd 171 | . . . . . 6 | |
19 | 18 | elrab3 2869 | . . . . 5 |
20 | 17, 19 | ax-mp 5 | . . . 4 |
21 | 20 | biimpi 119 | . . 3 |
22 | ordtriexmidlem2 4480 | . . . 4 | |
23 | 22 | eqcoms 2160 | . . 3 |
24 | 21, 23 | orim12i 749 | . 2 |
25 | 15, 24 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 104 wo 698 wceq 1335 wcel 2128 wral 2435 crab 2439 cin 3101 c0 3394 cpw 3543 csn 3560 con0 4324 csuc 4326 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-uni 3774 df-tr 4064 df-iord 4327 df-on 4329 df-suc 4332 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |