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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 3182 | . . . . 5 | |
2 | p0ex 4112 | . . . . . 6 | |
3 | 2 | prid2 3630 | . . . . 5 |
4 | 2ordpr 4439 | . . . . . . 7 | |
5 | pp0ex 4113 | . . . . . . . 8 | |
6 | 5 | elon 4296 | . . . . . . 7 |
7 | 4, 6 | mpbir 145 | . . . . . 6 |
8 | ordtriexmidlem 4435 | . . . . . . . 8 | |
9 | ontr2exmid.1 | . . . . . . . 8 | |
10 | sseq1 3120 | . . . . . . . . . . . . 13 | |
11 | 10 | anbi1d 460 | . . . . . . . . . . . 12 |
12 | eleq1 2202 | . . . . . . . . . . . 12 | |
13 | 11, 12 | imbi12d 233 | . . . . . . . . . . 11 |
14 | 13 | ralbidv 2437 | . . . . . . . . . 10 |
15 | 14 | albidv 1796 | . . . . . . . . 9 |
16 | 15 | rspcv 2785 | . . . . . . . 8 |
17 | 8, 9, 16 | mp2 16 | . . . . . . 7 |
18 | sseq2 3121 | . . . . . . . . . . 11 | |
19 | eleq1 2202 | . . . . . . . . . . 11 | |
20 | 18, 19 | anbi12d 464 | . . . . . . . . . 10 |
21 | 20 | imbi1d 230 | . . . . . . . . 9 |
22 | 21 | ralbidv 2437 | . . . . . . . 8 |
23 | 2, 22 | spcv 2779 | . . . . . . 7 |
24 | 17, 23 | ax-mp 5 | . . . . . 6 |
25 | eleq2 2203 | . . . . . . . . 9 | |
26 | 25 | anbi2d 459 | . . . . . . . 8 |
27 | eleq2 2203 | . . . . . . . 8 | |
28 | 26, 27 | imbi12d 233 | . . . . . . 7 |
29 | 28 | rspcv 2785 | . . . . . 6 |
30 | 7, 24, 29 | mp2 16 | . . . . 5 |
31 | 1, 3, 30 | mp2an 422 | . . . 4 |
32 | elpri 3550 | . . . 4 | |
33 | 31, 32 | ax-mp 5 | . . 3 |
34 | ordtriexmidlem2 4436 | . . . 4 | |
35 | 0ex 4055 | . . . . 5 | |
36 | biidd 171 | . . . . 5 | |
37 | 35, 36 | rabsnt 3598 | . . . 4 |
38 | 34, 37 | orim12i 748 | . . 3 |
39 | 33, 38 | ax-mp 5 | . 2 |
40 | orcom 717 | . 2 | |
41 | 39, 40 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 wal 1329 wceq 1331 wcel 1480 wral 2416 crab 2420 wss 3071 c0 3363 csn 3527 cpr 3528 word 4284 con0 4285 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 |
This theorem is referenced by: (None) |
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