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Theorem t1sep2 19738
Description: Any two points in a T1 space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
t1sep.1  |-  X  = 
U. J
Assertion
Ref Expression
t1sep2  |-  ( ( J  e.  Fre  /\  A  e.  X  /\  B  e.  X )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B ) )
Distinct variable groups:    A, o    B, o    o, J    o, X

Proof of Theorem t1sep2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 t1top 19699 . . . . . 6  |-  ( J  e.  Fre  ->  J  e.  Top )
2 t1sep.1 . . . . . . 7  |-  X  = 
U. J
32toptopon 19303 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
41, 3sylib 196 . . . . 5  |-  ( J  e.  Fre  ->  J  e.  (TopOn `  X )
)
5 ist1-2 19716 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Fre  <->  A. x  e.  X  A. y  e.  X  ( A. o  e.  J  ( x  e.  o  ->  y  e.  o )  ->  x  =  y ) ) )
64, 5syl 16 . . . 4  |-  ( J  e.  Fre  ->  ( J  e.  Fre  <->  A. x  e.  X  A. y  e.  X  ( A. o  e.  J  (
x  e.  o  -> 
y  e.  o )  ->  x  =  y ) ) )
76ibi 241 . . 3  |-  ( J  e.  Fre  ->  A. x  e.  X  A. y  e.  X  ( A. o  e.  J  (
x  e.  o  -> 
y  e.  o )  ->  x  =  y ) )
8 eleq1 2539 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  o  <->  A  e.  o ) )
98imbi1d 317 . . . . . 6  |-  ( x  =  A  ->  (
( x  e.  o  ->  y  e.  o )  <->  ( A  e.  o  ->  y  e.  o ) ) )
109ralbidv 2906 . . . . 5  |-  ( x  =  A  ->  ( A. o  e.  J  ( x  e.  o  ->  y  e.  o )  <->  A. o  e.  J  ( A  e.  o  ->  y  e.  o ) ) )
11 eqeq1 2471 . . . . 5  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
1210, 11imbi12d 320 . . . 4  |-  ( x  =  A  ->  (
( A. o  e.  J  ( x  e.  o  ->  y  e.  o )  ->  x  =  y )  <->  ( A. o  e.  J  ( A  e.  o  ->  y  e.  o )  ->  A  =  y )
) )
13 eleq1 2539 . . . . . . 7  |-  ( y  =  B  ->  (
y  e.  o  <->  B  e.  o ) )
1413imbi2d 316 . . . . . 6  |-  ( y  =  B  ->  (
( A  e.  o  ->  y  e.  o )  <->  ( A  e.  o  ->  B  e.  o ) ) )
1514ralbidv 2906 . . . . 5  |-  ( y  =  B  ->  ( A. o  e.  J  ( A  e.  o  ->  y  e.  o )  <->  A. o  e.  J  ( A  e.  o  ->  B  e.  o ) ) )
16 eqeq2 2482 . . . . 5  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
1715, 16imbi12d 320 . . . 4  |-  ( y  =  B  ->  (
( A. o  e.  J  ( A  e.  o  ->  y  e.  o )  ->  A  =  y )  <->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B )
) )
1812, 17rspc2v 3228 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( A. o  e.  J  ( x  e.  o  ->  y  e.  o )  ->  x  =  y )  -> 
( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B ) ) )
197, 18mpan9 469 . 2  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B ) )
20193impb 1192 1  |-  ( ( J  e.  Fre  /\  A  e.  X  /\  B  e.  X )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   U.cuni 4251   ` cfv 5594   Topctop 19263  TopOnctopon 19264   Frect1 19676
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-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-rab 2826  df-v 3120  df-sbc 3337  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-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-iota 5557  df-fun 5596  df-fv 5602  df-topgen 14716  df-top 19268  df-topon 19271  df-cld 19388  df-t1 19683
This theorem is referenced by:  t1sep  19739  isr0  20106
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