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Theorem hauspwpwdom 20357
 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 5882 . . 3
21a1i 11 . 2
3 haustop 19700 . . . . . 6
4 hauspwpwf1.x . . . . . . 7
54topopn 19284 . . . . . 6
63, 5syl 16 . . . . 5
76adantr 465 . . . 4
8 simpr 461 . . . 4
97, 8ssexd 4600 . . 3
10 pwexg 4637 . . 3
11 pwexg 4637 . . 3
129, 10, 113syl 20 . 2
13 eqid 2467 . . 3
144, 13hauspwpwf1 20356 . 2
15 f1dom2g 7545 . 2
162, 12, 14, 15syl3anc 1228 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767  cab 2452  wrex 2818  cvv 3118   cin 3480   wss 3481  cpw 4016  cuni 4251   class class class wbr 4453   cmpt 4511  wf1 5591  cfv 5594   cdom 7526  ctop 19263  ccl 19387  cha 19677 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-dom 7530  df-top 19268  df-cld 19388  df-ntr 19389  df-cls 19390  df-haus 19684 This theorem is referenced by: (None)
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