Theorem indiscld 23074

Description: The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
indiscld	⊢ (Clsd‘{∅, 𝐴}) = {∅, 𝐴}

Proof of Theorem indiscld

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		indistop 22985	. . . . 5 ⊢ {∅, 𝐴} ∈ Top
2		indisuni 22986	. . . . . 6 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
3	2	iscld 23010	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → (𝑥 ∈ (Clsd‘{∅, 𝐴}) ↔ (𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})))
4	1, 3	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ (Clsd‘{∅, 𝐴}) ↔ (𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}))
5		dfss4 4197	. . . . . 6 ⊢ (𝑥 ⊆ ( I ‘𝐴) ↔ (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
6	5	birani 504	. . . . 5 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
7		simpr 485	. . . . . . 7 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})
8		indislem 22983	. . . . . . 7 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
9	7, 8	eleqtrrdi 2850	. . . . . 6 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)})
10		elpri 4579	. . . . . 6 ⊢ ((( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)} → ((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)))
11		difeq2 4051	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ∅))
12		dif0 4306	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ∅) = ( I ‘𝐴)
13	11, 12	eqtrdi 2790	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ( I ‘𝐴))
14		fvex 6840	. . . . . . . . . 10 ⊢ ( I ‘𝐴) ∈ V
15	14	prid2 4695	. . . . . . . . 9 ⊢ ( I ‘𝐴) ∈ {∅, ( I ‘𝐴)}
16	15, 8	eleqtri 2837	. . . . . . . 8 ⊢ ( I ‘𝐴) ∈ {∅, 𝐴}
17	13, 16	eqeltrdi 2847	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
18		difeq2 4051	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ( I ‘𝐴)))
19		difid 4304	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ( I ‘𝐴)) = ∅
20	18, 19	eqtrdi 2790	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ∅)
21		0ex 5229	. . . . . . . . 9 ⊢ ∅ ∈ V
22	21	prid1 4694	. . . . . . . 8 ⊢ ∅ ∈ {∅, 𝐴}
23	20, 22	eqeltrdi 2847	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
24	17, 23	jaoi 863	. . . . . 6 ⊢ (((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
25	9, 10, 24	3syl 18	. . . . 5 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
26	6, 25	eqeltrrd 2840	. . . 4 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
27	4, 26	sylbi 218	. . 3 ⊢ (𝑥 ∈ (Clsd‘{∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
28	27	ssriv 3919	. 2 ⊢ (Clsd‘{∅, 𝐴}) ⊆ {∅, 𝐴}
29		0cld 23021	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ∅ ∈ (Clsd‘{∅, 𝐴}))
30	1, 29	ax-mp 5	. . . 4 ⊢ ∅ ∈ (Clsd‘{∅, 𝐴})
31	2	topcld 23018	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴}))
32	1, 31	ax-mp 5	. . . 4 ⊢ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})
33		prssi 4752	. . . 4 ⊢ ((∅ ∈ (Clsd‘{∅, 𝐴}) ∧ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})) → {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴}))
34	30, 32, 33	mp2an 698	. . 3 ⊢ {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴})
35	8, 34	eqsstrri 3962	. 2 ⊢ {∅, 𝐴} ⊆ (Clsd‘{∅, 𝐴})
36	28, 35	eqssi 3931	1 ⊢ (Clsd‘{∅, 𝐴}) = {∅, 𝐴}

Syntax hints:   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4261  {cpr 4557   I cid 5512  ‘cfv 6485  Topctop 22876  Clsdccld 22999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-top 22877  df-topon 22894  df-cld 23002

This theorem is referenced by: (None)