Theorem indistps2 18734

Description: The indiscrete topology on a set A expressed as a topological space, using direct component assignments. Compare with indistps 18733. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 18735 and indistps2ALT 18736 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 4522	. . . 4 |- (/) e. _V
4		fvex 5801	. . . . 5 |- ( Base ` K ) e. _V
5	1, 4	eqeltrri 2536	. . . 4 |- A e. _V
6	3, 5	unipr 4204	. . 3 |- U. { (/) , A } = ( (/) u. A )
7		uncom 3600	. . 3 |- ( (/) u. A ) = ( A u. (/) )
8		un0 3762	. . 3 |- ( A u. (/) ) = A
9	6, 7, 8	3eqtrri 2485	. 2 |- A = U. { (/) , A }
10		indistop 18724	. 2 |- { (/) , A } e. Top
11	1, 2, 9, 10	istpsi 18667	1 |- K e. TopSp

Syntax hints:   = wceq 1370   e. wcel 1758   _Vcvv 3070   u. cun 3426   (/)c0 3737   {cpr 3979   U.cuni 4191   ` cfv 5518   Basecbs 14278   TopOpenctopn 14464   TopSpctps 18619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-top 18621  df-topon 18624  df-topsp 18625

This theorem is referenced by: (None)