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Theorem dfcon2OLD 25402
Description: An alternate definition of connectedness. (Moved into main set.mm as dfcon2 17201 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  |-  X  = 
U. J
Assertion
Ref Expression
dfcon2OLD  |-  ( J  e.  Top  ->  ( J  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( (
x  =/=  (/)  /\  y  =/=  (/)  /\  ( x  i^i  y )  =  (/) )  ->  X  =/=  ( x  u.  y
) ) ) )
Distinct variable groups:    x, y, J    x, X, y

Proof of Theorem dfcon2OLD
StepHypRef Expression
1 dfcon2OLD.1 . . . 4  |-  X  = 
U. J
21toptopon 16727 . . 3  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3 dfcon2 17201 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( ( x  =/=  (/)  /\  y  =/=  (/)  /\  (
x  i^i  y )  =  (/) )  ->  (
x  u.  y )  =/=  X ) ) )
42, 3sylbi 187 . 2  |-  ( J  e.  Top  ->  ( J  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( (
x  =/=  (/)  /\  y  =/=  (/)  /\  ( x  i^i  y )  =  (/) )  ->  ( x  u.  y )  =/= 
X ) ) )
5 necom 2560 . . . 4  |-  ( ( x  u.  y )  =/=  X  <->  X  =/=  ( x  u.  y
) )
65imbi2i 303 . . 3  |-  ( ( ( x  =/=  (/)  /\  y  =/=  (/)  /\  ( x  i^i  y )  =  (/) )  ->  ( x  u.  y )  =/= 
X )  <->  ( (
x  =/=  (/)  /\  y  =/=  (/)  /\  ( x  i^i  y )  =  (/) )  ->  X  =/=  ( x  u.  y
) ) )
762ralbii 2603 . 2  |-  ( A. x  e.  J  A. y  e.  J  (
( x  =/=  (/)  /\  y  =/=  (/)  /\  ( x  i^i  y )  =  (/) )  ->  ( x  u.  y )  =/= 
X )  <->  A. x  e.  J  A. y  e.  J  ( (
x  =/=  (/)  /\  y  =/=  (/)  /\  ( x  i^i  y )  =  (/) )  ->  X  =/=  ( x  u.  y
) ) )
84, 7syl6bb 252 1  |-  ( J  e.  Top  ->  ( J  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( (
x  =/=  (/)  /\  y  =/=  (/)  /\  ( x  i^i  y )  =  (/) )  ->  X  =/=  ( x  u.  y
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577    u. cun 3184    i^i cin 3185   (/)c0 3489   U.cuni 3864   ` cfv 5292   Topctop 16687  TopOnctopon 16688   Conccon 17193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300  df-top 16692  df-topon 16695  df-cld 16812  df-con 17194
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