Theorem indistopon 23328

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 4789	. . . . 5 ⊢ (𝑥 ⊆ {∅, 𝐴} ↔ ((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})))
2		unieq 4877	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
3		uni0 4895	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
4		0ex 5260	. . . . . . . . . . 11 ⊢ ∅ ∈ V
5	4	prid1 4719	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, 𝐴}
6	3, 5	eqeltri 2856	. . . . . . . . 9 ⊢ ∪ ∅ ∈ {∅, 𝐴}
7	2, 6	eqeltrdi 2868	. . . . . . . 8 ⊢ (𝑥 = ∅ → ∪ 𝑥 ∈ {∅, 𝐴})
8	7	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = ∅ → ∪ 𝑥 ∈ {∅, 𝐴}))
9		unieq 4877	. . . . . . . . 9 ⊢ (𝑥 = {∅} → ∪ 𝑥 = ∪ {∅})
10	4	unisn 4885	. . . . . . . . . 10 ⊢ ∪ {∅} = ∅
11	10, 5	eqeltri 2856	. . . . . . . . 9 ⊢ ∪ {∅} ∈ {∅, 𝐴}
12	9, 11	eqeltrdi 2868	. . . . . . . 8 ⊢ (𝑥 = {∅} → ∪ 𝑥 ∈ {∅, 𝐴})
13	12	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = {∅} → ∪ 𝑥 ∈ {∅, 𝐴}))
14	8, 13	jaod 873	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → ∪ 𝑥 ∈ {∅, 𝐴}))
15		unieq 4877	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴} → ∪ 𝑥 = ∪ {𝐴})
16		unisng 4884	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
17	15, 16	sylan9eqr 2817	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {𝐴}) → ∪ 𝑥 = 𝐴)
18		prid2g 4718	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {∅, 𝐴})
19	18	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {𝐴}) → 𝐴 ∈ {∅, 𝐴})
20	17, 19	eqeltrd 2860	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {𝐴}) → ∪ 𝑥 ∈ {∅, 𝐴})
21	20	ex 418	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = {𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
22		unieq 4877	. . . . . . . . . 10 ⊢ (𝑥 = {∅, 𝐴} → ∪ 𝑥 = ∪ {∅, 𝐴})
23		uniprg 4882	. . . . . . . . . . . 12 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
24	4, 23	mpan 703	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
25		uncom 4156	. . . . . . . . . . . 12 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
26		un0 4343	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ ∅) = 𝐴
27	25, 26	eqtri 2783	. . . . . . . . . . 11 ⊢ (∅ ∪ 𝐴) = 𝐴
28	24, 27	eqtrdi 2811	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ∪ {∅, 𝐴} = 𝐴)
29	22, 28	sylan9eqr 2817	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {∅, 𝐴}) → ∪ 𝑥 = 𝐴)
30	18	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {∅, 𝐴}) → 𝐴 ∈ {∅, 𝐴})
31	29, 30	eqeltrd 2860	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {∅, 𝐴}) → ∪ 𝑥 ∈ {∅, 𝐴})
32	31	ex 418	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
33	21, 32	jaod 873	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴}) → ∪ 𝑥 ∈ {∅, 𝐴}))
34	14, 33	jaod 873	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})) → ∪ 𝑥 ∈ {∅, 𝐴}))
35	1, 34	biimtrid 245	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
36	35	alrimiv 1959	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥(𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
37		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
38	37	elpr 4605	. . . . 5 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
39		vex 3454	. . . . . . . . 9 ⊢ 𝑦 ∈ V
40	39	elpr 4605	. . . . . . . 8 ⊢ (𝑦 ∈ {∅, 𝐴} ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴))
41		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑦 = ∅)
42	41	ineq2d 4116	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = (𝑥 ∩ ∅))
43		in0 4344	. . . . . . . . . . . . 13 ⊢ (𝑥 ∩ ∅) = ∅
44	42, 43	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = ∅)
45	44, 5	eqeltrdi 2868	. . . . . . . . . . 11 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
46	45	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
47		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → 𝑦 = ∅)
48	47	ineq2d 4116	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = (𝑥 ∩ ∅))
49	48, 43	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = ∅)
50	49, 5	eqeltrdi 2868	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
51	50	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
52		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → 𝑥 = ∅)
53	52	ineq1d 4115	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (∅ ∩ 𝑦))
54		0in 4346	. . . . . . . . . . . . 13 ⊢ (∅ ∩ 𝑦) = ∅
55	53, 54	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = ∅)
56	55, 5	eqeltrdi 2868	. . . . . . . . . . 11 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
57	56	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
58		ineq12 4111	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐴))
59	58	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐴))
60		inidm 4122	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ 𝐴) = 𝐴
61	59, 60	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → (𝑥 ∩ 𝑦) = 𝐴)
62	18	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → 𝐴 ∈ {∅, 𝐴})
63	61, 62	eqeltrd 2860	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
64	63	ex 418	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
65	46, 51, 57, 64	ccased 1054	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 = ∅ ∨ 𝑥 = 𝐴) ∧ (𝑦 = ∅ ∨ 𝑦 = 𝐴)) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
66	65	expdimp 458	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ((𝑦 = ∅ ∨ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
67	40, 66	biimtrid 245	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → (𝑦 ∈ {∅, 𝐴} → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
68	67	ralrimiv 3153	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
69	68	ex 418	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → ∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
70	38, 69	biimtrid 245	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {∅, 𝐴} → ∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
71	70	ralrimiv 3153	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
72		prex 5396	. . . 4 ⊢ {∅, 𝐴} ∈ V
73		istopg 23222	. . . 4 ⊢ ({∅, 𝐴} ∈ V → ({∅, 𝐴} ∈ Top ↔ (∀𝑥(𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}) ∧ ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})))
74	72, 73	mp1i 14	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({∅, 𝐴} ∈ Top ↔ (∀𝑥(𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}) ∧ ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})))
75	36, 71, 74	mpbir2and 726	. 2 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ Top)
76	28	eqcomd 2766	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = ∪ {∅, 𝐴})
77		istopon 23239	. 2 ⊢ ({∅, 𝐴} ∈ (TopOn‘𝐴) ↔ ({∅, 𝐴} ∈ Top ∧ 𝐴 = ∪ {∅, 𝐴}))
78	75, 76, 77	sylanbrc 595	1 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   ∪ cun 4148  ∅c0 4277  {csn 4584  {cpr 4586  ∪ cuni 4866  ‘cfv 6543  Topctop 23220  TopOnctopon 23237

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6497  df-fun 6545  df-fv 6551  df-top 23221  df-topon 23238

This theorem is used by:  indistop  23329  indisuni  23330  indistpsx  23337  indistpsALT  23340  indistps2ALT  23341  cnindis  23619  indishmph  24126  indistgp  24428  topdifinf  38409