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Theorem hauspwpwdom 18012
 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 5734 . . 3
21a1i 11 . 2
3 haustop 17387 . . . . . 6
4 hauspwpwf1.x . . . . . . 7
54topopn 16971 . . . . . 6
63, 5syl 16 . . . . 5
76adantr 452 . . . 4
8 simpr 448 . . . 4
97, 8ssexd 4342 . . 3
10 pwexg 4375 . . 3
11 pwexg 4375 . . 3
129, 10, 113syl 19 . 2
13 eqid 2435 . . 3
144, 13hauspwpwf1 18011 . 2
15 f1dom2g 7117 . 2
162, 12, 14, 15syl3anc 1184 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cab 2421  wrex 2698  cvv 2948   cin 3311   wss 3312  cpw 3791  cuni 4007   class class class wbr 4204   cmpt 4258  wf1 5443  cfv 5446   cdom 7099  ctop 16950  ccl 17074  cha 17364 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  ax-un 4693 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-int 4043  df-iun 4087  df-iin 4088  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-dom 7103  df-top 16955  df-cld 17075  df-ntr 17076  df-cls 17077  df-haus 17371
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