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Theorem t1sep2 17425
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 17386 . . . . . 6  |-  ( J  e.  Fre  ->  J  e.  Top )
2 t1sep.1 . . . . . . 7  |-  X  = 
U. J
32toptopon 16990 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
41, 3sylib 189 . . . . 5  |-  ( J  e.  Fre  ->  J  e.  (TopOn `  X )
)
5 ist1-2 17403 . . . . 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 233 . . 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 2495 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  o  <->  A  e.  o ) )
98imbi1d 309 . . . . . 6  |-  ( x  =  A  ->  (
( x  e.  o  ->  y  e.  o )  <->  ( A  e.  o  ->  y  e.  o ) ) )
109ralbidv 2717 . . . . 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 2441 . . . . 5  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
1210, 11imbi12d 312 . . . 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 2495 . . . . . . 7  |-  ( y  =  B  ->  (
y  e.  o  <->  B  e.  o ) )
1413imbi2d 308 . . . . . 6  |-  ( y  =  B  ->  (
( A  e.  o  ->  y  e.  o )  <->  ( A  e.  o  ->  B  e.  o ) ) )
1514ralbidv 2717 . . . . 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 2444 . . . . 5  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
1715, 16imbi12d 312 . . . 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 3050 . . 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 456 . 2  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B ) )
20193impb 1149 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   U.cuni 4007   ` cfv 5446   Topctop 16950  TopOnctopon 16951   Frect1 17363
This theorem is referenced by:  t1sep  17426  isr0  17761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-topgen 13659  df-top 16955  df-topon 16958  df-cld 17075  df-t1 17370
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