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Mirrors > Home > ILE Home > Th. List > ontr2exmid | Unicode version |
Description: An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
Ref | Expression |
---|---|
ontr2exmid.1 |
Ref | Expression |
---|---|
ontr2exmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3025 | . . . . 5 | |
2 | p0ex 3939 | . . . . . 6 | |
3 | 2 | prid2 3477 | . . . . 5 |
4 | 2ordpr 4249 | . . . . . . 7 | |
5 | pp0ex 3940 | . . . . . . . 8 | |
6 | 5 | elon 4111 | . . . . . . 7 |
7 | 4, 6 | mpbir 134 | . . . . . 6 |
8 | ordtriexmidlem 4245 | . . . . . . . 8 | |
9 | ontr2exmid.1 | . . . . . . . 8 | |
10 | sseq1 2966 | . . . . . . . . . . . . 13 | |
11 | 10 | anbi1d 438 | . . . . . . . . . . . 12 |
12 | eleq1 2100 | . . . . . . . . . . . 12 | |
13 | 11, 12 | imbi12d 223 | . . . . . . . . . . 11 |
14 | 13 | ralbidv 2326 | . . . . . . . . . 10 |
15 | 14 | albidv 1705 | . . . . . . . . 9 |
16 | 15 | rspcv 2652 | . . . . . . . 8 |
17 | 8, 9, 16 | mp2 16 | . . . . . . 7 |
18 | sseq2 2967 | . . . . . . . . . . 11 | |
19 | eleq1 2100 | . . . . . . . . . . 11 | |
20 | 18, 19 | anbi12d 442 | . . . . . . . . . 10 |
21 | 20 | imbi1d 220 | . . . . . . . . 9 |
22 | 21 | ralbidv 2326 | . . . . . . . 8 |
23 | 2, 22 | spcv 2646 | . . . . . . 7 |
24 | 17, 23 | ax-mp 7 | . . . . . 6 |
25 | eleq2 2101 | . . . . . . . . 9 | |
26 | 25 | anbi2d 437 | . . . . . . . 8 |
27 | eleq2 2101 | . . . . . . . 8 | |
28 | 26, 27 | imbi12d 223 | . . . . . . 7 |
29 | 28 | rspcv 2652 | . . . . . 6 |
30 | 7, 24, 29 | mp2 16 | . . . . 5 |
31 | 1, 3, 30 | mp2an 402 | . . . 4 |
32 | elpri 3398 | . . . 4 | |
33 | 31, 32 | ax-mp 7 | . . 3 |
34 | ordtriexmidlem2 4246 | . . . 4 | |
35 | 0ex 3884 | . . . . 5 | |
36 | biidd 161 | . . . . 5 | |
37 | 35, 36 | rabsnt 3445 | . . . 4 |
38 | 34, 37 | orim12i 676 | . . 3 |
39 | 33, 38 | ax-mp 7 | . 2 |
40 | orcom 647 | . 2 | |
41 | 39, 40 | mpbi 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 wal 1241 wceq 1243 wcel 1393 wral 2306 crab 2310 wss 2917 c0 3224 csn 3375 cpr 3376 word 4099 con0 4100 |
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-in1 544 ax-in2 545 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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 |
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-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 |
This theorem is referenced by: (None) |
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