Theorem indistopon 22151

Description: The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
indistopon	⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))

Proof of Theorem indistopon

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sspr 4766	. . . . 5 ⊢ (𝑥 ⊆ {∅, 𝐴} ↔ ((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})))
2		unieq 4850	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
3		uni0 4869	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
4		0ex 5231	. . . . . . . . . . 11 ⊢ ∅ ∈ V
5	4	prid1 4698	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, 𝐴}
6	3, 5	eqeltri 2835	. . . . . . . . 9 ⊢ ∪ ∅ ∈ {∅, 𝐴}
7	2, 6	eqeltrdi 2847	. . . . . . . 8 ⊢ (𝑥 = ∅ → ∪ 𝑥 ∈ {∅, 𝐴})
8	7	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = ∅ → ∪ 𝑥 ∈ {∅, 𝐴}))
9		unieq 4850	. . . . . . . . 9 ⊢ (𝑥 = {∅} → ∪ 𝑥 = ∪ {∅})
10	4	unisn 4861	. . . . . . . . . 10 ⊢ ∪ {∅} = ∅
11	10, 5	eqeltri 2835	. . . . . . . . 9 ⊢ ∪ {∅} ∈ {∅, 𝐴}
12	9, 11	eqeltrdi 2847	. . . . . . . 8 ⊢ (𝑥 = {∅} → ∪ 𝑥 ∈ {∅, 𝐴})
13	12	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = {∅} → ∪ 𝑥 ∈ {∅, 𝐴}))
14	8, 13	jaod 856	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → ∪ 𝑥 ∈ {∅, 𝐴}))
15		unieq 4850	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴} → ∪ 𝑥 = ∪ {𝐴})
16		unisng 4860	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
17	15, 16	sylan9eqr 2800	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {𝐴}) → ∪ 𝑥 = 𝐴)
18		prid2g 4697	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {∅, 𝐴})
19	18	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {𝐴}) → 𝐴 ∈ {∅, 𝐴})
20	17, 19	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {𝐴}) → ∪ 𝑥 ∈ {∅, 𝐴})
21	20	ex 413	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = {𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
22		unieq 4850	. . . . . . . . . 10 ⊢ (𝑥 = {∅, 𝐴} → ∪ 𝑥 = ∪ {∅, 𝐴})
23		uniprg 4856	. . . . . . . . . . . 12 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
24	4, 23	mpan 687	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
25		uncom 4087	. . . . . . . . . . . 12 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
26		un0 4324	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ ∅) = 𝐴
27	25, 26	eqtri 2766	. . . . . . . . . . 11 ⊢ (∅ ∪ 𝐴) = 𝐴
28	24, 27	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ∪ {∅, 𝐴} = 𝐴)
29	22, 28	sylan9eqr 2800	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {∅, 𝐴}) → ∪ 𝑥 = 𝐴)
30	18	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {∅, 𝐴}) → 𝐴 ∈ {∅, 𝐴})
31	29, 30	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {∅, 𝐴}) → ∪ 𝑥 ∈ {∅, 𝐴})
32	31	ex 413	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
33	21, 32	jaod 856	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴}) → ∪ 𝑥 ∈ {∅, 𝐴}))
34	14, 33	jaod 856	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})) → ∪ 𝑥 ∈ {∅, 𝐴}))
35	1, 34	syl5bi 241	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
36	35	alrimiv 1930	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥(𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
37		vex 3436	. . . . . 6 ⊢ 𝑥 ∈ V
38	37	elpr 4584	. . . . 5 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
39		vex 3436	. . . . . . . . 9 ⊢ 𝑦 ∈ V
40	39	elpr 4584	. . . . . . . 8 ⊢ (𝑦 ∈ {∅, 𝐴} ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴))
41		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑦 = ∅)
42	41	ineq2d 4146	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = (𝑥 ∩ ∅))
43		in0 4325	. . . . . . . . . . . . 13 ⊢ (𝑥 ∩ ∅) = ∅
44	42, 43	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = ∅)
45	44, 5	eqeltrdi 2847	. . . . . . . . . . 11 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
46	45	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
47		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → 𝑦 = ∅)
48	47	ineq2d 4146	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = (𝑥 ∩ ∅))
49	48, 43	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = ∅)
50	49, 5	eqeltrdi 2847	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
51	50	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
52		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → 𝑥 = ∅)
53	52	ineq1d 4145	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (∅ ∩ 𝑦))
54		0in 4327	. . . . . . . . . . . . 13 ⊢ (∅ ∩ 𝑦) = ∅
55	53, 54	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = ∅)
56	55, 5	eqeltrdi 2847	. . . . . . . . . . 11 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
57	56	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
58		ineq12 4141	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐴))
59	58	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐴))
60		inidm 4152	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ 𝐴) = 𝐴
61	59, 60	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → (𝑥 ∩ 𝑦) = 𝐴)
62	18	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → 𝐴 ∈ {∅, 𝐴})
63	61, 62	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
64	63	ex 413	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
65	46, 51, 57, 64	ccased 1036	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 = ∅ ∨ 𝑥 = 𝐴) ∧ (𝑦 = ∅ ∨ 𝑦 = 𝐴)) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
66	65	expdimp 453	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ((𝑦 = ∅ ∨ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
67	40, 66	syl5bi 241	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → (𝑦 ∈ {∅, 𝐴} → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
68	67	ralrimiv 3102	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
69	68	ex 413	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → ∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
70	38, 69	syl5bi 241	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {∅, 𝐴} → ∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
71	70	ralrimiv 3102	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
72		prex 5355	. . . 4 ⊢ {∅, 𝐴} ∈ V
73		istopg 22044	. . . 4 ⊢ ({∅, 𝐴} ∈ V → ({∅, 𝐴} ∈ Top ↔ (∀𝑥(𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}) ∧ ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})))
74	72, 73	mp1i 13	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({∅, 𝐴} ∈ Top ↔ (∀𝑥(𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}) ∧ ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})))
75	36, 71, 74	mpbir2and 710	. 2 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ Top)
76	28	eqcomd 2744	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = ∪ {∅, 𝐴})
77		istopon 22061	. 2 ⊢ ({∅, 𝐴} ∈ (TopOn‘𝐴) ↔ ({∅, 𝐴} ∈ Top ∧ 𝐴 = ∪ {∅, 𝐴}))
78	75, 76, 77	sylanbrc 583	1 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561  {cpr 4563  ∪ cuni 4839  ‘cfv 6433  Topctop 22042  TopOnctopon 22059

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-top 22043  df-topon 22060

This theorem is referenced by:  indistop  22152  indisuni  22153  indistpsx  22160  indistpsALT  22163  indistpsALTOLD  22164  indistps2ALT  22165  cnindis  22443  indishmph  22949  indistgp  23251  topdifinf  35520