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Theorem dfcon2OLD 26253
 Description: An alternate definition of connectedness. (Moved into main set.mm as dfcon2 17145 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 8-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dfcon2OLD.1
Assertion
Ref Expression
dfcon2OLD
Distinct variable groups:   ,,   ,,

Proof of Theorem dfcon2OLD
StepHypRef Expression
1 dfcon2OLD.1 . . . 4
21toptopon 16671 . . 3 TopOn
3 dfcon2 17145 . . 3 TopOn
42, 3sylbi 187 . 2
5 necom 2527 . . . 4
65imbi2i 303 . . 3
762ralbii 2569 . 2
84, 7syl6bb 252 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   w3a 934   wceq 1623   wcel 1684   wne 2446  wral 2543   cun 3150   cin 3151  c0 3455  cuni 3827  cfv 5255  ctop 16631  TopOnctopon 16632  ccon 17137 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-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-rab 2552  df-v 2790  df-sbc 2992  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-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-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-top 16636  df-topon 16639  df-cld 16756  df-con 17138
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