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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 2296 | . . 3 | |
2 | eleq1 2233 | . . . . . 6 | |
3 | 2 | anbi1d 462 | . . . . 5 |
4 | 3 | simprbda 381 | . . . 4 |
5 | 4 | exlimiv 1591 | . . 3 |
6 | 1, 5 | sylbi 120 | . 2 |
7 | df-rab 2457 | . 2 | |
8 | 6, 7 | eleq2s 2265 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wex 1485 wcel 2141 cab 2156 crab 2452 |
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-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-rab 2457 |
This theorem is referenced by: ordtriexmidlem 4501 ordtri2or2exmidlem 4508 onsucelsucexmidlem 4511 ordsoexmid 4544 reg3exmidlemwe 4561 elfvmptrab1 5588 acexmidlemcase 5846 ssfirab 6908 exmidonfinlem 7159 cc4f 7220 genpelvl 7463 genpelvu 7464 suplocsrlempr 7758 nnindnn 7844 sup3exmid 8862 nnind 8883 supinfneg 9543 infsupneg 9544 supminfex 9545 ublbneg 9561 hashinfuni 10700 zsupcllemstep 11889 infssuzex 11893 infssuzledc 11894 bezoutlemsup 11953 uzwodc 11981 lcmgcdlem 12020 phisum 12183 oddennn 12336 evenennn 12337 znnen 12342 ennnfonelemg 12347 txdis1cn 13033 reopnap 13293 divcnap 13310 limccl 13383 dvlemap 13404 dvaddxxbr 13420 dvmulxxbr 13421 dvcoapbr 13426 dvcjbr 13427 dvrecap 13432 dveflem 13442 |
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