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Mirrors > Home > ILE Home > Th. List > nlimsucg | Unicode version |
Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
nlimsucg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limord 4317 | . . . . . 6 | |
2 | ordsuc 4478 | . . . . . 6 | |
3 | 1, 2 | sylibr 133 | . . . . 5 |
4 | limuni 4318 | . . . . 5 | |
5 | 3, 4 | jca 304 | . . . 4 |
6 | ordtr 4300 | . . . . . . . 8 | |
7 | unisucg 4336 | . . . . . . . . 9 | |
8 | 7 | biimpa 294 | . . . . . . . 8 |
9 | 6, 8 | sylan2 284 | . . . . . . 7 |
10 | 9 | eqeq2d 2151 | . . . . . 6 |
11 | ordirr 4457 | . . . . . . . . 9 | |
12 | eleq2 2203 | . . . . . . . . . 10 | |
13 | 12 | notbid 656 | . . . . . . . . 9 |
14 | 11, 13 | syl5ibrcom 156 | . . . . . . . 8 |
15 | sucidg 4338 | . . . . . . . . 9 | |
16 | 15 | con3i 621 | . . . . . . . 8 |
17 | 14, 16 | syl6 33 | . . . . . . 7 |
18 | 17 | adantl 275 | . . . . . 6 |
19 | 10, 18 | sylbid 149 | . . . . 5 |
20 | 19 | expimpd 360 | . . . 4 |
21 | 5, 20 | syl5 32 | . . 3 |
22 | 21 | con2d 613 | . 2 |
23 | 22 | pm2.43i 49 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1331 wcel 1480 cuni 3736 wtr 4026 word 4284 wlim 4286 csuc 4287 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-setind 4452 |
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-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-uni 3737 df-tr 4027 df-iord 4288 df-ilim 4291 df-suc 4293 |
This theorem is referenced by: (None) |
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