Theorem indiscld 23229

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 23140	. . . . 5 ⊢ {∅, 𝐴} ∈ Top
2		indisuni 23141	. . . . . 6 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
3	2	iscld 23165	. . . . 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 4223	. . . . . 6 ⊢ (𝑥 ⊆ ( I ‘𝐴) ↔ (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
6	5	birani 508	. . . . 5 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
7		simpr 489	. . . . . . 7 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})
8		indislem 23138	. . . . . . 7 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
9	7, 8	eleqtrrdi 2874	. . . . . 6 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)})
10		elpri 4614	. . . . . 6 ⊢ ((( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)} → ((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)))
11		difeq2 4076	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ∅))
12		dif0 4335	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ∅) = ( I ‘𝐴)
13	11, 12	eqtrdi 2814	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ( I ‘𝐴))
14		fvex 6896	. . . . . . . . . 10 ⊢ ( I ‘𝐴) ∈ V
15	14	prid2 4730	. . . . . . . . 9 ⊢ ( I ‘𝐴) ∈ {∅, ( I ‘𝐴)}
16	15, 8	eleqtri 2861	. . . . . . . 8 ⊢ ( I ‘𝐴) ∈ {∅, 𝐴}
17	13, 16	eqeltrdi 2871	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
18		difeq2 4076	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ( I ‘𝐴)))
19		difid 4333	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ( I ‘𝐴)) = ∅
20	18, 19	eqtrdi 2814	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ∅)
21		0ex 5271	. . . . . . . . 9 ⊢ ∅ ∈ V
22	21	prid1 4729	. . . . . . . 8 ⊢ ∅ ∈ {∅, 𝐴}
23	20, 22	eqeltrdi 2871	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
24	17, 23	jaoi 870	. . . . . 6 ⊢ (((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
25	9, 10, 24	3syl 19	. . . . 5 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
26	6, 25	eqeltrrd 2864	. . . 4 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
27	4, 26	sylbi 220	. . 3 ⊢ (𝑥 ∈ (Clsd‘{∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
28	27	ssriv 3942	. 2 ⊢ (Clsd‘{∅, 𝐴}) ⊆ {∅, 𝐴}
29		0cld 23176	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ∅ ∈ (Clsd‘{∅, 𝐴}))
30	1, 29	ax-mp 5	. . . 4 ⊢ ∅ ∈ (Clsd‘{∅, 𝐴})
31	2	topcld 23173	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴}))
32	1, 31	ax-mp 5	. . . 4 ⊢ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})
33		prssi 4788	. . . 4 ⊢ ((∅ ∈ (Clsd‘{∅, 𝐴}) ∧ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})) → {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴}))
34	30, 32, 33	mp2an 704	. . 3 ⊢ {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴})
35	8, 34	eqsstrri 3985	. 2 ⊢ {∅, 𝐴} ⊆ (Clsd‘{∅, 𝐴})
36	28, 35	eqssi 3954	1 ⊢ (Clsd‘{∅, 𝐴}) = {∅, 𝐴}

Syntax hints:   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287  {cpr 4592   I cid 5557  ‘cfv 6538  Topctop 23031  Clsdccld 23154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-top 23032  df-topon 23049  df-cld 23157

This theorem is referenced by: (None)