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Theorem hauspwpwdom 17683
 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 5539 . . 3
21a1i 10 . 2
3 simpr 447 . . . 4
4 haustop 17059 . . . . . 6
5 hauspwpwf1.x . . . . . . 7
65topopn 16652 . . . . . 6
74, 6syl 15 . . . . 5
87adantr 451 . . . 4
9 ssexg 4160 . . . 4
103, 8, 9syl2anc 642 . . 3
11 pwexg 4194 . . 3
12 pwexg 4194 . . 3
1310, 11, 123syl 18 . 2
14 eqid 2283 . . 3
155, 14hauspwpwf1 17682 . 2
16 f1dom2g 6879 . 2
172, 13, 15, 16syl3anc 1182 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684  cab 2269  wrex 2544  cvv 2788   cin 3151   wss 3152  cpw 3625  cuni 3827   class class class wbr 4023   cmpt 4077  wf1 5252  cfv 5255   cdom 6861  ctop 16631  ccl 16755  cha 17036 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dom 6865  df-top 16636  df-cld 16756  df-ntr 16757  df-cls 16758  df-haus 17043
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