Theorem indiscld 22587

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 22497	. . . . 5 ⊢ {∅, 𝐴} ∈ Top
2		indisuni 22498	. . . . . 6 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
3	2	iscld 22523	. . . . 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 484	. . . . . 6 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ⊆ ( I ‘𝐴))
6		dfss4 4258	. . . . . 6 ⊢ (𝑥 ⊆ ( I ‘𝐴) ↔ (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
7	5, 6	sylib 217	. . . . 5 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
8		simpr 486	. . . . . . 7 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})
9		indislem 22495	. . . . . . 7 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
10	8, 9	eleqtrrdi 2845	. . . . . 6 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)})
11		elpri 4650	. . . . . 6 ⊢ ((( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)} → ((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)))
12		difeq2 4116	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ∅))
13		dif0 4372	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ∅) = ( I ‘𝐴)
14	12, 13	eqtrdi 2789	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ( I ‘𝐴))
15		fvex 6902	. . . . . . . . . 10 ⊢ ( I ‘𝐴) ∈ V
16	15	prid2 4767	. . . . . . . . 9 ⊢ ( I ‘𝐴) ∈ {∅, ( I ‘𝐴)}
17	16, 9	eleqtri 2832	. . . . . . . 8 ⊢ ( I ‘𝐴) ∈ {∅, 𝐴}
18	14, 17	eqeltrdi 2842	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
19		difeq2 4116	. . . . . . . . 9 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ( I ‘𝐴)))
20		difid 4370	. . . . . . . . 9 ⊢ (( I ‘𝐴) ∖ ( I ‘𝐴)) = ∅
21	19, 20	eqtrdi 2789	. . . . . . . 8 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ∅)
22		0ex 5307	. . . . . . . . 9 ⊢ ∅ ∈ V
23	22	prid1 4766	. . . . . . . 8 ⊢ ∅ ∈ {∅, 𝐴}
24	21, 23	eqeltrdi 2842	. . . . . . 7 ⊢ ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
25	18, 24	jaoi 856	. . . . . 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 2835	. . . 4 ⊢ ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
28	4, 27	sylbi 216	. . 3 ⊢ (𝑥 ∈ (Clsd‘{∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
29	28	ssriv 3986	. 2 ⊢ (Clsd‘{∅, 𝐴}) ⊆ {∅, 𝐴}
30		0cld 22534	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ∅ ∈ (Clsd‘{∅, 𝐴}))
31	1, 30	ax-mp 5	. . . 4 ⊢ ∅ ∈ (Clsd‘{∅, 𝐴})
32	2	topcld 22531	. . . . 5 ⊢ ({∅, 𝐴} ∈ Top → ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴}))
33	1, 32	ax-mp 5	. . . 4 ⊢ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})
34		prssi 4824	. . . 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 4017	. 2 ⊢ {∅, 𝐴} ⊆ (Clsd‘{∅, 𝐴})
37	29, 36	eqssi 3998	1 ⊢ (Clsd‘{∅, 𝐴}) = {∅, 𝐴}

Syntax hints:   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322  {cpr 4630   I cid 5573  ‘cfv 6541  Topctop 22387  Clsdccld 22512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6493  df-fun 6543  df-fv 6549  df-top 22388  df-topon 22405  df-cld 22515

This theorem is referenced by: (None)