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Mirrors > Home > ILE Home > Th. List > suctr | Unicode version |
Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) |
Ref | Expression |
---|---|
suctr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 |
. . . . 5
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2 | vex 2692 |
. . . . . 6
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3 | 2 | elsuc 4336 |
. . . . 5
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4 | 1, 3 | sylib 121 |
. . . 4
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5 | simpl 108 |
. . . . . . 7
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6 | eleq2 2204 |
. . . . . . 7
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7 | 5, 6 | syl5ibcom 154 |
. . . . . 6
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8 | elelsuc 4339 |
. . . . . 6
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9 | 7, 8 | syl6 33 |
. . . . 5
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10 | trel 4041 |
. . . . . . . . 9
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11 | 10 | expd 256 |
. . . . . . . 8
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12 | 11 | adantrd 277 |
. . . . . . 7
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13 | 12, 8 | syl8 71 |
. . . . . 6
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14 | jao 745 |
. . . . . 6
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15 | 13, 14 | syl6 33 |
. . . . 5
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16 | 9, 15 | mpdi 43 |
. . . 4
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17 | 4, 16 | mpdi 43 |
. . 3
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18 | 17 | alrimivv 1848 |
. 2
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19 | dftr2 4036 |
. 2
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20 | 18, 19 | sylibr 133 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-uni 3745 df-tr 4035 df-suc 4301 |
This theorem is referenced by: ordsucim 4424 ordom 4528 |
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