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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 4350 |
. 2
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2 | bi2.04 248 |
. . . . . . 7
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3 | 2 | albii 1481 |
. . . . . 6
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4 | df-ral 2473 |
. . . . . 6
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5 | 3, 4 | bitr4i 187 |
. . . . 5
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6 | sbim 1965 |
. . . . . . . . . . 11
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7 | clelsb1 2294 |
. . . . . . . . . . . 12
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8 | clelsb1 2294 |
. . . . . . . . . . . 12
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9 | 7, 8 | imbi12i 239 |
. . . . . . . . . . 11
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10 | 6, 9 | bitri 184 |
. . . . . . . . . 10
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11 | 10 | ralbii 2496 |
. . . . . . . . 9
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12 | ralcom3 2658 |
. . . . . . . . 9
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13 | 11, 12 | bitri 184 |
. . . . . . . 8
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14 | epel 4310 |
. . . . . . . . . 10
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15 | 14 | imbi1i 238 |
. . . . . . . . 9
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16 | 15 | ralbii 2496 |
. . . . . . . 8
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17 | 13, 16 | bitr4i 187 |
. . . . . . 7
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18 | 17 | imbi1i 238 |
. . . . . 6
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19 | 18 | ralbii 2496 |
. . . . 5
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20 | 5, 19 | bitri 184 |
. . . 4
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21 | ax-setind 4554 |
. . . . 5
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22 | dfss2 3159 |
. . . . 5
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23 | 21, 22 | sylibr 134 |
. . . 4
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24 | 20, 23 | sylbir 135 |
. . 3
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25 | df-frfor 4349 |
. . 3
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26 | 24, 25 | mpbir 146 |
. 2
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27 | 1, 26 | mpgbir 1464 |
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-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-eprel 4307 df-frfor 4349 df-frind 4350 |
This theorem is referenced by: ordfr 4592 wessep 4595 reg3exmidlemwe 4596 |
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