Theorem indiscld 23066

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 22977	. . . . 5 ⊢ {∅, 𝐴} ∈ Top
2		indisuni 22978	. . . . . 6 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
3	2	iscld 23002	. . . . 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		simpl 482	. . . . . 6 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ⊆ ( I ‘𝐴))
6		dfss4 4210	. . . . . 6 ⊢ (𝑥 ⊆ ( I ‘𝐴) ↔ (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
7	5, 6	sylib 218	. . . . 5 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
8		simpr 484	. . . . . . 7 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})
9		indislem 22975	. . . . . . 7 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
10	8, 9	eleqtrrdi 2848	. . . . . 6 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)})
11		elpri 4592	. . . . . 6 ⊢ ((( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)} → ((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)))
12		difeq2 4061	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ∅))
13		dif0 4319	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ∅) = ( I ‘𝐴)
14	12, 13	eqtrdi 2788	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ( I ‘𝐴))
15		fvex 6847	. . . . . . . . . 10 ⊢ ( I ‘𝐴) ∈ V
16	15	prid2 4708	. . . . . . . . 9 ⊢ ( I ‘𝐴) ∈ {∅, ( I ‘𝐴)}
17	16, 9	eleqtri 2835	. . . . . . . 8 ⊢ ( I ‘𝐴) ∈ {∅, 𝐴}
18	14, 17	eqeltrdi 2845	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
19		difeq2 4061	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ( I ‘𝐴)))
20		difid 4317	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ( I ‘𝐴)) = ∅
21	19, 20	eqtrdi 2788	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ∅)
22		0ex 5242	. . . . . . . . 9 ⊢ ∅ ∈ V
23	22	prid1 4707	. . . . . . . 8 ⊢ ∅ ∈ {∅, 𝐴}
24	21, 23	eqeltrdi 2845	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
25	18, 24	jaoi 858	. . . . . 6 ⊢ (((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
26	10, 11, 25	3syl 18	. . . . 5 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
27	7, 26	eqeltrrd 2838	. . . 4 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
28	4, 27	sylbi 217	. . 3 ⊢ (𝑥 ∈ (Clsd‘{∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
29	28	ssriv 3926	. 2 ⊢ (Clsd‘{∅, 𝐴}) ⊆ {∅, 𝐴}
30		0cld 23013	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ∅ ∈ (Clsd‘{∅, 𝐴}))
31	1, 30	ax-mp 5	. . . 4 ⊢ ∅ ∈ (Clsd‘{∅, 𝐴})
32	2	topcld 23010	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴}))
33	1, 32	ax-mp 5	. . . 4 ⊢ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})
34		prssi 4765	. . . 4 ⊢ ((∅ ∈ (Clsd‘{∅, 𝐴}) ∧ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})) → {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴}))
35	31, 33, 34	mp2an 693	. . 3 ⊢ {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴})
36	9, 35	eqsstrri 3970	. 2 ⊢ {∅, 𝐴} ⊆ (Clsd‘{∅, 𝐴})
37	29, 36	eqssi 3939	1 ⊢ (Clsd‘{∅, 𝐴}) = {∅, 𝐴}

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274  {cpr 4570   I cid 5518  ‘cfv 6492  Topctop 22868  Clsdccld 22991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-top 22869  df-topon 22886  df-cld 22994

This theorem is referenced by: (None)