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Mirrors > Home > ILE Home > Th. List > sucprcreg | Unicode version |
Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.) |
Ref | Expression |
---|---|
sucprcreg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucprc 4430 |
. 2
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2 | elirr 4558 |
. . . 4
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3 | nfv 1539 |
. . . . 5
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4 | eleq1 2252 |
. . . . 5
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5 | 3, 4 | ceqsalg 2780 |
. . . 4
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6 | 2, 5 | mtbiri 676 |
. . 3
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7 | velsn 3624 |
. . . . 5
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8 | olc 712 |
. . . . . 6
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9 | elun 3291 |
. . . . . . 7
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10 | ssid 3190 |
. . . . . . . . 9
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11 | df-suc 4389 |
. . . . . . . . . . 11
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12 | 11 | eqeq1i 2197 |
. . . . . . . . . 10
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13 | sseq1 3193 |
. . . . . . . . . 10
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14 | 12, 13 | sylbi 121 |
. . . . . . . . 9
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15 | 10, 14 | mpbiri 168 |
. . . . . . . 8
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16 | 15 | sseld 3169 |
. . . . . . 7
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17 | 9, 16 | biimtrrid 153 |
. . . . . 6
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18 | 8, 17 | syl5 32 |
. . . . 5
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19 | 7, 18 | biimtrrid 153 |
. . . 4
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20 | 19 | alrimiv 1885 |
. . 3
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21 | 6, 20 | nsyl3 627 |
. 2
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22 | 1, 21 | impbii 126 |
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 615 ax-in2 616 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 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-sn 3613 df-suc 4389 |
This theorem is referenced by: (None) |
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