Theorem indistps2 19379

Description: The indiscrete topology on a set A expressed as a topological space, using direct component assignments. Compare with indistps 19378. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 19380 and indistps2ALT 19381 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)

Hypotheses

Ref	Expression
indistps2.a	|- ( Base ` K ) = A
indistps2.j	|- ( TopOpen ` K ) = { (/) , A }

Assertion

Ref	Expression
indistps2	|- K e. TopSp

Proof of Theorem indistps2

Step	Hyp	Ref	Expression
1		indistps2.a	. 2 |- ( Base ` K ) = A
2		indistps2.j	. 2 |- ( TopOpen ` K ) = { (/) , A }
3		0ex 4563	. . . 4 |- (/) e. _V
4		fvex 5862	. . . . 5 |- ( Base ` K ) e. _V
5	1, 4	eqeltrri 2526	. . . 4 |- A e. _V
6	3, 5	unipr 4243	. . 3 |- U. { (/) , A } = ( (/) u. A )
7		uncom 3630	. . 3 |- ( (/) u. A ) = ( A u. (/) )
8		un0 3792	. . 3 |- ( A u. (/) ) = A
9	6, 7, 8	3eqtrri 2475	. 2 |- A = U. { (/) , A }
10		indistop 19369	. 2 |- { (/) , A } e. Top
11	1, 2, 9, 10	istpsi 19312	1 |- K e. TopSp

Syntax hints:   = wceq 1381   e. wcel 1802   _Vcvv 3093   u. cun 3456   (/)c0 3767   {cpr 4012   U.cuni 4230   ` cfv 5574   Basecbs 14504   TopOpenctopn 14691   TopSpctps 19264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-iota 5537  df-fun 5576  df-fv 5582  df-top 19266  df-topon 19269  df-topsp 19270

This theorem is referenced by: (None)