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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-frind 4254 | . 2 FrFor | |
2 | bi2.04 247 | . . . . . . 7 | |
3 | 2 | albii 1446 | . . . . . 6 |
4 | df-ral 2421 | . . . . . 6 | |
5 | 3, 4 | bitr4i 186 | . . . . 5 |
6 | sbim 1926 | . . . . . . . . . . 11 | |
7 | clelsb3 2244 | . . . . . . . . . . . 12 | |
8 | clelsb3 2244 | . . . . . . . . . . . 12 | |
9 | 7, 8 | imbi12i 238 | . . . . . . . . . . 11 |
10 | 6, 9 | bitri 183 | . . . . . . . . . 10 |
11 | 10 | ralbii 2441 | . . . . . . . . 9 |
12 | ralcom3 2598 | . . . . . . . . 9 | |
13 | 11, 12 | bitri 183 | . . . . . . . 8 |
14 | epel 4214 | . . . . . . . . . 10 | |
15 | 14 | imbi1i 237 | . . . . . . . . 9 |
16 | 15 | ralbii 2441 | . . . . . . . 8 |
17 | 13, 16 | bitr4i 186 | . . . . . . 7 |
18 | 17 | imbi1i 237 | . . . . . 6 |
19 | 18 | ralbii 2441 | . . . . 5 |
20 | 5, 19 | bitri 183 | . . . 4 |
21 | ax-setind 4452 | . . . . 5 | |
22 | dfss2 3086 | . . . . 5 | |
23 | 21, 22 | sylibr 133 | . . . 4 |
24 | 20, 23 | sylbir 134 | . . 3 |
25 | df-frfor 4253 | . . 3 FrFor | |
26 | 24, 25 | mpbir 145 | . 2 FrFor |
27 | 1, 26 | mpgbir 1429 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wal 1329 wcel 1480 wsb 1735 wral 2416 wss 3071 class class class wbr 3929 cep 4209 FrFor wfrfor 4249 wfr 4250 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-eprel 4211 df-frfor 4253 df-frind 4254 |
This theorem is referenced by: ordfr 4489 wessep 4492 reg3exmidlemwe 4493 |
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