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Theorem hausnei 19956
 Description: Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
Hypothesis
Ref Expression
ist0.1
Assertion
Ref Expression
hausnei
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem hausnei
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ist0.1 . . . . . . 7
21ishaus 19950 . . . . . 6
32simprbi 464 . . . . 5
4 neeq1 2738 . . . . . . 7
5 eleq1 2529 . . . . . . . . 9
653anbi1d 1303 . . . . . . . 8
762rexbidv 2975 . . . . . . 7
84, 7imbi12d 320 . . . . . 6
9 neeq2 2740 . . . . . . 7
10 eleq1 2529 . . . . . . . . 9
11103anbi2d 1304 . . . . . . . 8
12112rexbidv 2975 . . . . . . 7
139, 12imbi12d 320 . . . . . 6
148, 13rspc2v 3219 . . . . 5
153, 14syl5 32 . . . 4
1615ex 434 . . 3
1716com3r 79 . 2
18173imp2 1211 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1395   wcel 1819   wne 2652  wral 2807  wrex 2808   cin 3470  c0 3793  cuni 4251  ctop 19521  cha 19936 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-uni 4252  df-haus 19943 This theorem is referenced by:  haust1  19980  cnhaus  19982  lmmo  20008  hauscmplem  20033  pthaus  20265  txhaus  20274  xkohaus  20280  hausflimi  20607  hauspwpwf1  20614
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