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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 4367 | . . . . . 6 | |
2 | ordsuc 4534 | . . . . . 6 | |
3 | 1, 2 | sylibr 133 | . . . . 5 |
4 | limuni 4368 | . . . . 5 | |
5 | 3, 4 | jca 304 | . . . 4 |
6 | ordtr 4350 | . . . . . . . 8 | |
7 | unisucg 4386 | . . . . . . . . 9 | |
8 | 7 | biimpa 294 | . . . . . . . 8 |
9 | 6, 8 | sylan2 284 | . . . . . . 7 |
10 | 9 | eqeq2d 2176 | . . . . . 6 |
11 | ordirr 4513 | . . . . . . . . 9 | |
12 | eleq2 2228 | . . . . . . . . . 10 | |
13 | 12 | notbid 657 | . . . . . . . . 9 |
14 | 11, 13 | syl5ibrcom 156 | . . . . . . . 8 |
15 | sucidg 4388 | . . . . . . . . 9 | |
16 | 15 | con3i 622 | . . . . . . . 8 |
17 | 14, 16 | syl6 33 | . . . . . . 7 |
18 | 17 | adantl 275 | . . . . . 6 |
19 | 10, 18 | sylbid 149 | . . . . 5 |
20 | 19 | expimpd 361 | . . . 4 |
21 | 5, 20 | syl5 32 | . . 3 |
22 | 21 | con2d 614 | . 2 |
23 | 22 | pm2.43i 49 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1342 wcel 2135 cuni 3783 wtr 4074 word 4334 wlim 4336 csuc 4337 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-uni 3784 df-tr 4075 df-iord 4338 df-ilim 4341 df-suc 4343 |
This theorem is referenced by: (None) |
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