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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limord 4395 |
. . . . . 6
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2 | ordsuc 4562 |
. . . . . 6
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3 | 1, 2 | sylibr 134 |
. . . . 5
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4 | limuni 4396 |
. . . . 5
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5 | 3, 4 | jca 306 |
. . . 4
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6 | ordtr 4378 |
. . . . . . . 8
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7 | unisucg 4414 |
. . . . . . . . 9
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8 | 7 | biimpa 296 |
. . . . . . . 8
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9 | 6, 8 | sylan2 286 |
. . . . . . 7
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10 | 9 | eqeq2d 2189 |
. . . . . 6
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11 | ordirr 4541 |
. . . . . . . . 9
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12 | eleq2 2241 |
. . . . . . . . . 10
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13 | 12 | notbid 667 |
. . . . . . . . 9
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14 | 11, 13 | syl5ibrcom 157 |
. . . . . . . 8
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15 | sucidg 4416 |
. . . . . . . . 9
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16 | 15 | con3i 632 |
. . . . . . . 8
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17 | 14, 16 | syl6 33 |
. . . . . . 7
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18 | 17 | adantl 277 |
. . . . . 6
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19 | 10, 18 | sylbid 150 |
. . . . 5
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20 | 19 | expimpd 363 |
. . . 4
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21 | 5, 20 | syl5 32 |
. . 3
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22 | 21 | con2d 624 |
. 2
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23 | 22 | pm2.43i 49 |
1
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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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-setind 4536 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-uni 3810 df-tr 4102 df-iord 4366 df-ilim 4369 df-suc 4371 |
This theorem is referenced by: (None) |
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