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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clelab 2237 |
. . 3
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2 | eleq1 2175 |
. . . . . 6
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3 | 2 | anbi1d 458 |
. . . . 5
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4 | 3 | simprbda 378 |
. . . 4
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5 | 4 | exlimiv 1558 |
. . 3
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6 | 1, 5 | sylbi 120 |
. 2
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7 | df-rab 2397 |
. 2
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8 | 6, 7 | eleq2s 2207 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-rab 2397 |
This theorem is referenced by: ordtriexmidlem 4393 ordtri2or2exmidlem 4399 onsucelsucexmidlem 4402 ordsoexmid 4435 reg3exmidlemwe 4451 elfvmptrab1 5467 acexmidlemcase 5721 ssfirab 6771 genpelvl 7261 genpelvu 7262 nnindnn 7621 sup3exmid 8618 nnind 8639 supinfneg 9285 infsupneg 9286 supminfex 9287 ublbneg 9300 hashinfuni 10409 zsupcllemstep 11479 infssuzex 11483 infssuzledc 11484 bezoutlemsup 11536 lcmgcdlem 11597 oddennn 11743 evenennn 11744 znnen 11749 ennnfonelemg 11754 txdis1cn 12282 divcnap 12534 limccl 12577 dvlemap 12597 |
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