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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 4287 | . . . . . 6 | |
2 | ordsuc 4448 | . . . . . 6 | |
3 | 1, 2 | sylibr 133 | . . . . 5 |
4 | limuni 4288 | . . . . 5 | |
5 | 3, 4 | jca 304 | . . . 4 |
6 | ordtr 4270 | . . . . . . . 8 | |
7 | unisucg 4306 | . . . . . . . . 9 | |
8 | 7 | biimpa 294 | . . . . . . . 8 |
9 | 6, 8 | sylan2 284 | . . . . . . 7 |
10 | 9 | eqeq2d 2129 | . . . . . 6 |
11 | ordirr 4427 | . . . . . . . . 9 | |
12 | eleq2 2181 | . . . . . . . . . 10 | |
13 | 12 | notbid 641 | . . . . . . . . 9 |
14 | 11, 13 | syl5ibrcom 156 | . . . . . . . 8 |
15 | sucidg 4308 | . . . . . . . . 9 | |
16 | 15 | con3i 606 | . . . . . . . 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 598 | . 2 |
23 | 22 | pm2.43i 49 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1316 wcel 1465 cuni 3706 wtr 3996 word 4254 wlim 4256 csuc 4257 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-uni 3707 df-tr 3997 df-iord 4258 df-ilim 4261 df-suc 4263 |
This theorem is referenced by: (None) |
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