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Mirrors > Home > ILE Home > Th. List > ordsucunielexmid | Unicode version |
Description: The converse of sucunielr 4421 (where is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
ordsucunielexmid.1 |
Ref | Expression |
---|---|
ordsucunielexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4292 | . . . . . . . 8 | |
2 | ordtr 4295 | . . . . . . . 8 | |
3 | 1, 2 | syl 14 | . . . . . . 7 |
4 | vex 2684 | . . . . . . . 8 | |
5 | 4 | unisuc 4330 | . . . . . . 7 |
6 | 3, 5 | sylib 121 | . . . . . 6 |
7 | 6 | eleq2d 2207 | . . . . 5 |
8 | 7 | adantl 275 | . . . 4 |
9 | suceloni 4412 | . . . . 5 | |
10 | ordsucunielexmid.1 | . . . . . 6 | |
11 | eleq1 2200 | . . . . . . . 8 | |
12 | suceq 4319 | . . . . . . . . 9 | |
13 | 12 | eleq1d 2206 | . . . . . . . 8 |
14 | 11, 13 | imbi12d 233 | . . . . . . 7 |
15 | unieq 3740 | . . . . . . . . 9 | |
16 | 15 | eleq2d 2207 | . . . . . . . 8 |
17 | eleq2 2201 | . . . . . . . 8 | |
18 | 16, 17 | imbi12d 233 | . . . . . . 7 |
19 | 14, 18 | rspc2va 2798 | . . . . . 6 |
20 | 10, 19 | mpan2 421 | . . . . 5 |
21 | 9, 20 | sylan2 284 | . . . 4 |
22 | 8, 21 | sylbird 169 | . . 3 |
23 | 22 | rgen2a 2484 | . 2 |
24 | 23 | onsucelsucexmid 4440 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 wral 2414 cuni 3731 wtr 4021 word 4279 con0 4280 csuc 4282 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 |
This theorem is referenced by: (None) |
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