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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 4430 |
. . . . . 6
| |
| 2 | ordsuc 4599 |
. . . . . 6
| |
| 3 | 1, 2 | sylibr 134 |
. . . . 5
|
| 4 | limuni 4431 |
. . . . 5
| |
| 5 | 3, 4 | jca 306 |
. . . 4
|
| 6 | ordtr 4413 |
. . . . . . . 8
| |
| 7 | unisucg 4449 |
. . . . . . . . 9
| |
| 8 | 7 | biimpa 296 |
. . . . . . . 8
|
| 9 | 6, 8 | sylan2 286 |
. . . . . . 7
|
| 10 | 9 | eqeq2d 2208 |
. . . . . 6
|
| 11 | ordirr 4578 |
. . . . . . . . 9
| |
| 12 | eleq2 2260 |
. . . . . . . . . 10
| |
| 13 | 12 | notbid 668 |
. . . . . . . . 9
|
| 14 | 11, 13 | syl5ibrcom 157 |
. . . . . . . 8
|
| 15 | sucidg 4451 |
. . . . . . . . 9
| |
| 16 | 15 | con3i 633 |
. . . . . . . 8
|
| 17 | 14, 16 | syl6 33 |
. . . . . . 7
|
| 18 | 17 | adantl 277 |
. . . . . 6
|
| 19 | 10, 18 | sylbid 150 |
. . . . 5
|
| 20 | 19 | expimpd 363 |
. . . 4
|
| 21 | 5, 20 | syl5 32 |
. . 3
|
| 22 | 21 | con2d 625 |
. 2
|
| 23 | 22 | pm2.43i 49 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-ext 2178 ax-setind 4573 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-v 2765 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-sn 3628 df-pr 3629 df-uni 3840 df-tr 4132 df-iord 4401 df-ilim 4404 df-suc 4406 |
| This theorem is referenced by: (None) |
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