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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 4115 |
. 2
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2 | elirr 4224 |
. . . 4
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3 | nfv 1418 |
. . . . 5
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4 | eleq1 2097 |
. . . . 5
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5 | 3, 4 | ceqsalg 2576 |
. . . 4
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6 | 2, 5 | mtbiri 599 |
. . 3
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7 | elsn 3382 |
. . . . 5
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8 | olc 631 |
. . . . . 6
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9 | elun 3078 |
. . . . . . 7
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10 | ssid 2958 |
. . . . . . . . 9
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11 | df-suc 4074 |
. . . . . . . . . . 11
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12 | 11 | eqeq1i 2044 |
. . . . . . . . . 10
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13 | sseq1 2960 |
. . . . . . . . . 10
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14 | 12, 13 | sylbi 114 |
. . . . . . . . 9
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15 | 10, 14 | mpbiri 157 |
. . . . . . . 8
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16 | 15 | sseld 2938 |
. . . . . . 7
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17 | 9, 16 | syl5bir 142 |
. . . . . 6
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18 | 8, 17 | syl5 28 |
. . . . 5
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19 | 7, 18 | syl5bir 142 |
. . . 4
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20 | 19 | alrimiv 1751 |
. . 3
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21 | 6, 20 | nsyl3 556 |
. 2
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22 | 1, 21 | impbii 117 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-suc 4074 |
This theorem is referenced by: (None) |
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