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Theorem llynlly 17532
 Description: A locally space is n-locally : the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llynlly Locally 𝑛Locally

Proof of Theorem llynlly
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 17527 . 2 Locally
2 llyi 17529 . . . . 5 Locally t
3 simpl1 960 . . . . . . . . . . 11 Locally t Locally
43, 1syl 16 . . . . . . . . . 10 Locally t
5 simprl 733 . . . . . . . . . 10 Locally t
6 simprr2 1006 . . . . . . . . . 10 Locally t
7 opnneip 17175 . . . . . . . . . 10
84, 5, 6, 7syl3anc 1184 . . . . . . . . 9 Locally t
9 simprr1 1005 . . . . . . . . . 10 Locally t
10 vex 2951 . . . . . . . . . . 11
1110elpw 3797 . . . . . . . . . 10
129, 11sylibr 204 . . . . . . . . 9 Locally t
13 elin 3522 . . . . . . . . 9
148, 12, 13sylanbrc 646 . . . . . . . 8 Locally t
15 simprr3 1007 . . . . . . . 8 Locally t t
1614, 15jca 519 . . . . . . 7 Locally t t
1716ex 424 . . . . . 6 Locally t t
1817reximdv2 2807 . . . . 5 Locally t t
192, 18mpd 15 . . . 4 Locally t
20193expb 1154 . . 3 Locally t
2120ralrimivva 2790 . 2 Locally t
22 isnlly 17524 . 2 𝑛Locally t
231, 21, 22sylanbrc 646 1 Locally 𝑛Locally
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wcel 1725  wral 2697  wrex 2698   cin 3311   wss 3312  cpw 3791  csn 3806  cfv 5446  (class class class)co 6073   ↾t crest 13640  ctop 16950  cnei 17153  Locally clly 17519  𝑛Locally cnlly 17520 This theorem is referenced by:  llyssnlly  17533  symgtgp  18123 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-top 16955  df-nei 17154  df-lly 17521  df-nlly 17522
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