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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 4069 | . 2 FrFor | |
2 | bi2.04 237 | . . . . . . 7 | |
3 | 2 | albii 1359 | . . . . . 6 |
4 | df-ral 2311 | . . . . . 6 | |
5 | 3, 4 | bitr4i 176 | . . . . 5 |
6 | sbim 1827 | . . . . . . . . . . 11 | |
7 | clelsb3 2142 | . . . . . . . . . . . 12 | |
8 | clelsb3 2142 | . . . . . . . . . . . 12 | |
9 | 7, 8 | imbi12i 228 | . . . . . . . . . . 11 |
10 | 6, 9 | bitri 173 | . . . . . . . . . 10 |
11 | 10 | ralbii 2330 | . . . . . . . . 9 |
12 | ralcom3 2477 | . . . . . . . . 9 | |
13 | 11, 12 | bitri 173 | . . . . . . . 8 |
14 | epel 4029 | . . . . . . . . . 10 | |
15 | 14 | imbi1i 227 | . . . . . . . . 9 |
16 | 15 | ralbii 2330 | . . . . . . . 8 |
17 | 13, 16 | bitr4i 176 | . . . . . . 7 |
18 | 17 | imbi1i 227 | . . . . . 6 |
19 | 18 | ralbii 2330 | . . . . 5 |
20 | 5, 19 | bitri 173 | . . . 4 |
21 | ax-setind 4262 | . . . . 5 | |
22 | dfss2 2934 | . . . . 5 | |
23 | 21, 22 | sylibr 137 | . . . 4 |
24 | 20, 23 | sylbir 125 | . . 3 |
25 | df-frfor 4068 | . . 3 FrFor | |
26 | 24, 25 | mpbir 134 | . 2 FrFor |
27 | 1, 26 | mpgbir 1342 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wal 1241 wcel 1393 wsb 1645 wral 2306 wss 2917 class class class wbr 3764 cep 4024 FrFor wfrfor 4064 wfr 4065 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-eprel 4026 df-frfor 4068 df-frind 4069 |
This theorem is referenced by: ordfr 4299 wessep 4302 reg3exmidlemwe 4303 |
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