Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  t0sep Structured version   Unicode version

Theorem t0sep 19951
 Description: Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
ist0.1
Assertion
Ref Expression
t0sep
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem t0sep
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ist0.1 . . . 4
21ist0 19947 . . 3
32simprbi 464 . 2
4 eleq1 2529 . . . . . . 7
54bibi1d 319 . . . . . 6
65ralbidv 2896 . . . . 5
7 eqeq1 2461 . . . . 5
86, 7imbi12d 320 . . . 4
9 eleq1 2529 . . . . . . 7
109bibi2d 318 . . . . . 6
1110ralbidv 2896 . . . . 5
12 eqeq2 2472 . . . . 5
1311, 12imbi12d 320 . . . 4
148, 13rspc2va 3220 . . 3
1514ancoms 453 . 2
163, 15sylan 471 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  wral 2807  cuni 4251  ctop 19520  ct0 19933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-uni 4252  df-t0 19940 This theorem is referenced by:  t0dist  19952  cnt0  19973
 Copyright terms: Public domain W3C validator