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Mirrors > Home > ILE Home > Th. List > ordsucim | Unicode version |
Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
Ref | Expression |
---|---|
ordsucim |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 4399 |
. . 3
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2 | suctr 4442 |
. . 3
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3 | 1, 2 | syl 14 |
. 2
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4 | df-suc 4392 |
. . . . . 6
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5 | 4 | eleq2i 2256 |
. . . . 5
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6 | elun 3291 |
. . . . 5
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7 | velsn 3627 |
. . . . . 6
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8 | 7 | orbi2i 763 |
. . . . 5
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9 | 5, 6, 8 | 3bitri 206 |
. . . 4
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10 | dford3 4388 |
. . . . . . . 8
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11 | 10 | simprbi 275 |
. . . . . . 7
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12 | df-ral 2473 |
. . . . . . 7
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13 | 11, 12 | sylib 122 |
. . . . . 6
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14 | 13 | 19.21bi 1569 |
. . . . 5
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15 | treq 4125 |
. . . . . 6
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16 | 1, 15 | syl5ibrcom 157 |
. . . . 5
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17 | 14, 16 | jaod 718 |
. . . 4
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18 | 9, 17 | biimtrid 152 |
. . 3
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19 | 18 | ralrimiv 2562 |
. 2
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20 | dford3 4388 |
. 2
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21 | 3, 19, 20 | sylanbrc 417 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3616 df-uni 3828 df-tr 4120 df-iord 4387 df-suc 4392 |
This theorem is referenced by: onsuc 4521 ordsucg 4522 onsucsssucr 4529 ordtriexmidlem 4539 2ordpr 4544 ordsuc 4583 nnsucsssuc 6521 |
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