Theorem indiscld 21696

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 21607	. . . . 5 ⊢ {∅, 𝐴} ∈ Top
2		indisuni 21608	. . . . . 6 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
3	2	iscld 21632	. . . . 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 486	. . . . . 6 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ⊆ ( I ‘𝐴))
6		dfss4 4185	. . . . . 6 ⊢ (𝑥 ⊆ ( I ‘𝐴) ↔ (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
7	5, 6	sylib 221	. . . . 5 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
8		simpr 488	. . . . . . 7 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})
9		indislem 21605	. . . . . . 7 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
10	8, 9	eleqtrrdi 2901	. . . . . 6 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)})
11		elpri 4547	. . . . . 6 ⊢ ((( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)} → ((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)))
12		difeq2 4044	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ∅))
13		dif0 4286	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ∅) = ( I ‘𝐴)
14	12, 13	eqtrdi 2849	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ( I ‘𝐴))
15		fvex 6658	. . . . . . . . . 10 ⊢ ( I ‘𝐴) ∈ V
16	15	prid2 4659	. . . . . . . . 9 ⊢ ( I ‘𝐴) ∈ {∅, ( I ‘𝐴)}
17	16, 9	eleqtri 2888	. . . . . . . 8 ⊢ ( I ‘𝐴) ∈ {∅, 𝐴}
18	14, 17	eqeltrdi 2898	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
19		difeq2 4044	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ( I ‘𝐴)))
20		difid 4284	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ( I ‘𝐴)) = ∅
21	19, 20	eqtrdi 2849	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ∅)
22		0ex 5175	. . . . . . . . 9 ⊢ ∅ ∈ V
23	22	prid1 4658	. . . . . . . 8 ⊢ ∅ ∈ {∅, 𝐴}
24	21, 23	eqeltrdi 2898	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
25	18, 24	jaoi 854	. . . . . 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 2891	. . . 4 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
28	4, 27	sylbi 220	. . 3 ⊢ (𝑥 ∈ (Clsd‘{∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
29	28	ssriv 3919	. 2 ⊢ (Clsd‘{∅, 𝐴}) ⊆ {∅, 𝐴}
30		0cld 21643	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ∅ ∈ (Clsd‘{∅, 𝐴}))
31	1, 30	ax-mp 5	. . . 4 ⊢ ∅ ∈ (Clsd‘{∅, 𝐴})
32	2	topcld 21640	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴}))
33	1, 32	ax-mp 5	. . . 4 ⊢ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})
34		prssi 4714	. . . 4 ⊢ ((∅ ∈ (Clsd‘{∅, 𝐴}) ∧ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})) → {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴}))
35	31, 33, 34	mp2an 691	. . 3 ⊢ {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴})
36	9, 35	eqsstrri 3950	. 2 ⊢ {∅, 𝐴} ⊆ (Clsd‘{∅, 𝐴})
37	29, 36	eqssi 3931	1 ⊢ (Clsd‘{∅, 𝐴}) = {∅, 𝐴}

Syntax hints:   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  {cpr 4527   I cid 5424  ‘cfv 6324  Topctop 21498  Clsdccld 21621

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-top 21499  df-topon 21516  df-cld 21624

This theorem is referenced by: (None)