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Mirrors > Home > ILE Home > Th. List > limcrcl | Unicode version |
Description: Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.) |
Ref | Expression |
---|---|
limcrcl | lim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-limced 13121 | . . 3 lim # | |
2 | 1 | elmpocl 6021 | . 2 lim |
3 | cnex 7859 | . . . . 5 | |
4 | 3, 3 | elpm2 6628 | . . . 4 |
5 | 4 | anbi1i 454 | . . 3 |
6 | df-3an 965 | . . 3 | |
7 | 5, 6 | bitr4i 186 | . 2 |
8 | 2, 7 | sylib 121 | 1 lim |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wcel 2128 wral 2435 wrex 2436 crab 2439 wss 3102 class class class wbr 3967 cdm 4589 wf 5169 cfv 5173 (class class class)co 5827 cpm 6597 cc 7733 clt 7915 cmin 8051 # cap 8461 crp 9567 cabs 10909 lim climc 13119 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-fv 5181 df-ov 5830 df-oprab 5831 df-mpo 5832 df-pm 6599 df-limced 13121 |
This theorem is referenced by: limccl 13124 limcdifap 13127 limcimolemlt 13129 limcresi 13131 limccnpcntop 13140 limccnp2lem 13141 limccnp2cntop 13142 limccoap 13143 |
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