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Mirrors > Home > ILE Home > Th. List > ordtri2orexmid | Unicode version |
Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
Ref | Expression |
---|---|
ordtri2orexmid.1 |
Ref | Expression |
---|---|
ordtri2orexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtri2orexmid.1 | . . . 4 | |
2 | ordtriexmidlem 4503 | . . . . 5 | |
3 | suc0 4396 | . . . . . 6 | |
4 | 0elon 4377 | . . . . . . 7 | |
5 | 4 | onsuci 4500 | . . . . . 6 |
6 | 3, 5 | eqeltrri 2244 | . . . . 5 |
7 | eleq1 2233 | . . . . . . 7 | |
8 | sseq2 3171 | . . . . . . 7 | |
9 | 7, 8 | orbi12d 788 | . . . . . 6 |
10 | eleq2 2234 | . . . . . . 7 | |
11 | sseq1 3170 | . . . . . . 7 | |
12 | 10, 11 | orbi12d 788 | . . . . . 6 |
13 | 9, 12 | rspc2va 2848 | . . . . 5 |
14 | 2, 6, 13 | mpanl12 434 | . . . 4 |
15 | 1, 14 | ax-mp 5 | . . 3 |
16 | elsni 3601 | . . . . 5 | |
17 | ordtriexmidlem2 4504 | . . . . 5 | |
18 | 16, 17 | syl 14 | . . . 4 |
19 | snssg 3716 | . . . . . 6 | |
20 | 4, 19 | ax-mp 5 | . . . . 5 |
21 | 0ex 4116 | . . . . . . . 8 | |
22 | 21 | snid 3614 | . . . . . . 7 |
23 | biidd 171 | . . . . . . . 8 | |
24 | 23 | elrab3 2887 | . . . . . . 7 |
25 | 22, 24 | ax-mp 5 | . . . . . 6 |
26 | 25 | biimpi 119 | . . . . 5 |
27 | 20, 26 | sylbir 134 | . . . 4 |
28 | 18, 27 | orim12i 754 | . . 3 |
29 | 15, 28 | ax-mp 5 | . 2 |
30 | orcom 723 | . 2 | |
31 | 29, 30 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 104 wo 703 wceq 1348 wcel 2141 wral 2448 crab 2452 wss 3121 c0 3414 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-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 |
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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