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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 3232 | . . . . 5 | |
2 | p0ex 4174 | . . . . . 6 | |
3 | 2 | prid2 3690 | . . . . 5 |
4 | 2ordpr 4508 | . . . . . . 7 | |
5 | pp0ex 4175 | . . . . . . . 8 | |
6 | 5 | elon 4359 | . . . . . . 7 |
7 | 4, 6 | mpbir 145 | . . . . . 6 |
8 | ordtriexmidlem 4503 | . . . . . . . 8 | |
9 | ontr2exmid.1 | . . . . . . . 8 | |
10 | sseq1 3170 | . . . . . . . . . . . . 13 | |
11 | 10 | anbi1d 462 | . . . . . . . . . . . 12 |
12 | eleq1 2233 | . . . . . . . . . . . 12 | |
13 | 11, 12 | imbi12d 233 | . . . . . . . . . . 11 |
14 | 13 | ralbidv 2470 | . . . . . . . . . 10 |
15 | 14 | albidv 1817 | . . . . . . . . 9 |
16 | 15 | rspcv 2830 | . . . . . . . 8 |
17 | 8, 9, 16 | mp2 16 | . . . . . . 7 |
18 | sseq2 3171 | . . . . . . . . . . 11 | |
19 | eleq1 2233 | . . . . . . . . . . 11 | |
20 | 18, 19 | anbi12d 470 | . . . . . . . . . 10 |
21 | 20 | imbi1d 230 | . . . . . . . . 9 |
22 | 21 | ralbidv 2470 | . . . . . . . 8 |
23 | 2, 22 | spcv 2824 | . . . . . . 7 |
24 | 17, 23 | ax-mp 5 | . . . . . 6 |
25 | eleq2 2234 | . . . . . . . . 9 | |
26 | 25 | anbi2d 461 | . . . . . . . 8 |
27 | eleq2 2234 | . . . . . . . 8 | |
28 | 26, 27 | imbi12d 233 | . . . . . . 7 |
29 | 28 | rspcv 2830 | . . . . . 6 |
30 | 7, 24, 29 | mp2 16 | . . . . 5 |
31 | 1, 3, 30 | mp2an 424 | . . . 4 |
32 | elpri 3606 | . . . 4 | |
33 | 31, 32 | ax-mp 5 | . . 3 |
34 | ordtriexmidlem2 4504 | . . . 4 | |
35 | 0ex 4116 | . . . . 5 | |
36 | biidd 171 | . . . . 5 | |
37 | 35, 36 | rabsnt 3658 | . . . 4 |
38 | 34, 37 | orim12i 754 | . . 3 |
39 | 33, 38 | ax-mp 5 | . 2 |
40 | orcom 723 | . 2 | |
41 | 39, 40 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 wal 1346 wceq 1348 wcel 2141 wral 2448 crab 2452 wss 3121 c0 3414 csn 3583 cpr 3584 word 4347 con0 4348 |
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-pr 3590 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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