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Mirrors > Home > ILE Home > Th. List > wessep | Unicode version |
Description: A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.) |
Ref | Expression |
---|---|
wessep |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2939 | . . . . . . 7 | |
2 | ssel 2939 | . . . . . . 7 | |
3 | ssel 2939 | . . . . . . 7 | |
4 | 1, 2, 3 | 3anim123d 1214 | . . . . . 6 |
5 | 4 | adantl 262 | . . . . 5 |
6 | 5 | imdistani 419 | . . . 4 |
7 | wetrep 4097 | . . . . . 6 | |
8 | 7 | adantlr 446 | . . . . 5 |
9 | epel 4029 | . . . . . 6 | |
10 | epel 4029 | . . . . . 6 | |
11 | 9, 10 | anbi12i 433 | . . . . 5 |
12 | epel 4029 | . . . . 5 | |
13 | 8, 11, 12 | 3imtr4g 194 | . . . 4 |
14 | 6, 13 | syl 14 | . . 3 |
15 | 14 | ralrimivvva 2402 | . 2 |
16 | zfregfr 4298 | . . 3 | |
17 | df-wetr 4071 | . . 3 | |
18 | 16, 17 | mpbiran 847 | . 2 |
19 | 15, 18 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wcel 1393 wral 2306 wss 2917 class class class wbr 3764 cep 4024 wfr 4065 wwe 4067 |
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 df-wetr 4071 |
This theorem is referenced by: (None) |
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