Theorem t0dist 21129

Description: Any two distinct points in a T₀ space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)

Hypothesis

Ref	Expression
ist0.1	|- X = U. J

Assertion

Ref	Expression
t0dist	|- ( ( J e. Kol2 /\ ( A e. X /\ B e. X /\ A =/= B ) ) -> E. o e. J -. ( A e. o <-> B e. o ) )

Distinct variable groups:   A, o   B, o   o, J   o, X

Proof of Theorem t0dist

Step	Hyp	Ref	Expression
1		ist0.1	. . . . . 6 |- X = U. J
2	1	t0sep 21128	. . . . 5 |- ( ( J e. Kol2 /\ ( A e. X /\ B e. X ) ) -> ( A. o e. J ( A e. o <-> B e. o ) -> A = B ) )
3	2	necon3ad 2807	. . . 4 |- ( ( J e. Kol2 /\ ( A e. X /\ B e. X ) ) -> ( A =/= B -> -. A. o e. J ( A e. o <-> B e. o ) ) )
4	3	exp32 631	. . 3 |- ( J e. Kol2 -> ( A e. X -> ( B e. X -> ( A =/= B -> -. A. o e. J ( A e. o <-> B e. o ) ) ) ) )
5	4	3imp2 1282	. 2 |- ( ( J e. Kol2 /\ ( A e. X /\ B e. X /\ A =/= B ) ) -> -. A. o e. J ( A e. o <-> B e. o ) )
6		rexnal 2995	. 2 |- ( E. o e. J -. ( A e. o <-> B e. o ) <-> -. A. o e. J ( A e. o <-> B e. o ) )
7	5, 6	sylibr 224	1 |- ( ( J e. Kol2 /\ ( A e. X /\ B e. X /\ A =/= B ) ) -> E. o e. J -. ( A e. o <-> B e. o ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   U.cuni 4436   Kol2ct0 21110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-uni 4437  df-t0 21117

This theorem is referenced by: (None)