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Mirrors > Home > ILE Home > Th. List > zfregfr | Unicode version |
Description: The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
Ref | Expression |
---|---|
zfregfr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-frind 4183 |
. 2
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2 | bi2.04 247 |
. . . . . . 7
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3 | 2 | albii 1411 |
. . . . . 6
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4 | df-ral 2375 |
. . . . . 6
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5 | 3, 4 | bitr4i 186 |
. . . . 5
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6 | sbim 1882 |
. . . . . . . . . . 11
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7 | clelsb3 2199 |
. . . . . . . . . . . 12
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8 | clelsb3 2199 |
. . . . . . . . . . . 12
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9 | 7, 8 | imbi12i 238 |
. . . . . . . . . . 11
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10 | 6, 9 | bitri 183 |
. . . . . . . . . 10
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11 | 10 | ralbii 2395 |
. . . . . . . . 9
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12 | ralcom3 2548 |
. . . . . . . . 9
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13 | 11, 12 | bitri 183 |
. . . . . . . 8
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14 | epel 4143 |
. . . . . . . . . 10
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15 | 14 | imbi1i 237 |
. . . . . . . . 9
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16 | 15 | ralbii 2395 |
. . . . . . . 8
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17 | 13, 16 | bitr4i 186 |
. . . . . . 7
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18 | 17 | imbi1i 237 |
. . . . . 6
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19 | 18 | ralbii 2395 |
. . . . 5
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20 | 5, 19 | bitri 183 |
. . . 4
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21 | ax-setind 4381 |
. . . . 5
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22 | dfss2 3028 |
. . . . 5
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23 | 21, 22 | sylibr 133 |
. . . 4
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24 | 20, 23 | sylbir 134 |
. . 3
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25 | df-frfor 4182 |
. . 3
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26 | 24, 25 | mpbir 145 |
. 2
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27 | 1, 26 | mpgbir 1394 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-setind 4381 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-br 3868 df-opab 3922 df-eprel 4140 df-frfor 4182 df-frind 4183 |
This theorem is referenced by: ordfr 4418 wessep 4421 reg3exmidlemwe 4422 |
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