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Mirrors > Home > ILE Home > Th. List > elrabi | Unicode version |
Description: Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
Ref | Expression |
---|---|
elrabi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clelab 2203 | . . 3 | |
2 | eleq1 2141 | . . . . . 6 | |
3 | 2 | anbi1d 452 | . . . . 5 |
4 | 3 | simprbda 375 | . . . 4 |
5 | 4 | exlimiv 1529 | . . 3 |
6 | 1, 5 | sylbi 119 | . 2 |
7 | df-rab 2357 | . 2 | |
8 | 6, 7 | eleq2s 2173 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 cab 2067 crab 2352 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-rab 2357 |
This theorem is referenced by: ordtriexmidlem 4263 ordtri2or2exmidlem 4269 onsucelsucexmidlem 4272 ordsoexmid 4305 reg3exmidlemwe 4321 acexmidlemcase 5527 genpelvl 6702 genpelvu 6703 nnindnn 7059 nnind 8055 supinfneg 8683 infsupneg 8684 supminfex 8685 ublbneg 8698 zsupcllemstep 10341 infssuzex 10345 infssuzledc 10346 bezoutlemsup 10398 lcmgcdlem 10459 |
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