Theorem indistop 23030

Description: The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.) (Revised by Stefan Allan, 6-Nov-2008.) (Revised by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
indistop	⊢ {∅, 𝐴} ∈ Top

Proof of Theorem indistop

Step	Hyp	Ref	Expression
1		indislem 23028	. 2 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
2		fvex 6933	. . . 4 ⊢ ( I ‘𝐴) ∈ V
3		indistopon 23029	. . . 4 ⊢ (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)))
4	2, 3	ax-mp 5	. . 3 ⊢ {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))
5	4	topontopi 22942	. 2 ⊢ {∅, ( I ‘𝐴)} ∈ Top
6	1, 5	eqeltrri 2841	1 ⊢ {∅, 𝐴} ∈ Top

Syntax hints:   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  {cpr 4650   I cid 5592  ‘cfv 6573  Topctop 22920  TopOnctopon 22937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-top 22921  df-topon 22938

This theorem is referenced by:  indistpsx  23038  indistps  23039  indistps2  23040  indiscld  23120  indisconn  23447  txindis  23663  indispconn  35202  onpsstopbas  36396