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Theorem hauspwpwdom 19661
Description: If is a Hausdorff space, then the cardinality of the closure of a set is bounded by the double powerset of . In particular, a Hausdorff space with a dense subset has cardinality at most , and a separable Hausdorff space has cardinality at most . (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 28-Jul-2015.)
Hypothesis
Ref Expression
hauspwpwf1.x
Assertion
Ref Expression
hauspwpwdom

Proof of Theorem hauspwpwdom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5783 . . 3
21a1i 11 . 2
3 haustop 19035 . . . . . 6
4 hauspwpwf1.x . . . . . . 7
54topopn 18619 . . . . . 6
63, 5syl 16 . . . . 5
76adantr 465 . . . 4
8 simpr 461 . . . 4
97, 8ssexd 4521 . . 3
10 pwexg 4558 . . 3
11 pwexg 4558 . . 3
129, 10, 113syl 20 . 2
13 eqid 2450 . . 3
144, 13hauspwpwf1 19660 . 2
15 f1dom2g 7411 . 2
162, 12, 14, 15syl3anc 1219 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1370  e.wcel 1757  {cab 2435  E.wrex 2793   cvv 3052  i^icin 3409  C_wss 3410  ~Pcpw 3942  U.cuni 4173   class class class wbr 4374  e.cmpt 4432  -1-1->wf1 5497  `cfv 5500   cdom 7392   ctop 18598   ccl 18722   cha 19012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4485  ax-sep 4495  ax-nul 4503  ax-pow 4552  ax-pr 4613  ax-un 6456
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3054  df-sbc 3269  df-csb 3371  df-dif 3413  df-un 3415  df-in 3417  df-ss 3424  df-nul 3720  df-if 3874  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4174  df-int 4211  df-iun 4255  df-iin 4256  df-br 4375  df-opab 4433  df-mpt 4434  df-id 4718  df-xp 4928  df-rel 4929  df-cnv 4930  df-co 4931  df-dm 4932  df-rn 4933  df-res 4934  df-ima 4935  df-iota 5463  df-fun 5502  df-fn 5503  df-f 5504  df-f1 5505  df-fo 5506  df-f1o 5507  df-fv 5508  df-dom 7396  df-top 18603  df-cld 18723  df-ntr 18724  df-cls 18725  df-haus 19019
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