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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 3096 | . . . . . . 7 | |
2 | ssel 3096 | . . . . . . 7 | |
3 | ssel 3096 | . . . . . . 7 | |
4 | 1, 2, 3 | 3anim123d 1298 | . . . . . 6 |
5 | 4 | adantl 275 | . . . . 5 |
6 | 5 | imdistani 442 | . . . 4 |
7 | wetrep 4290 | . . . . . 6 | |
8 | 7 | adantlr 469 | . . . . 5 |
9 | epel 4222 | . . . . . 6 | |
10 | epel 4222 | . . . . . 6 | |
11 | 9, 10 | anbi12i 456 | . . . . 5 |
12 | epel 4222 | . . . . 5 | |
13 | 8, 11, 12 | 3imtr4g 204 | . . . 4 |
14 | 6, 13 | syl 14 | . . 3 |
15 | 14 | ralrimivvva 2518 | . 2 |
16 | zfregfr 4496 | . . 3 | |
17 | df-wetr 4264 | . . 3 | |
18 | 16, 17 | mpbiran 925 | . 2 |
19 | 15, 18 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wcel 1481 wral 2417 wss 3076 class class class wbr 3937 cep 4217 wfr 4258 wwe 4260 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-eprel 4219 df-frfor 4261 df-frind 4262 df-wetr 4264 |
This theorem is referenced by: (None) |
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