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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 2162 | . . 3 | |
2 | eleq1 2100 | . . . . . 6 | |
3 | 2 | anbi1d 438 | . . . . 5 |
4 | 3 | simprbda 365 | . . . 4 |
5 | 4 | exlimiv 1489 | . . 3 |
6 | 1, 5 | sylbi 114 | . 2 |
7 | df-rab 2315 | . 2 | |
8 | 6, 7 | eleq2s 2132 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cab 2026 crab 2310 |
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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-rab 2315 |
This theorem is referenced by: ordtriexmidlem 4245 ordtri2or2exmidlem 4251 onsucelsucexmidlem 4254 ordsoexmid 4286 reg3exmidlemwe 4303 acexmidlemcase 5507 genpelvl 6610 genpelvu 6611 nnindnn 6967 nnind 7930 ublbneg 8548 |
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