Theorem indistopon 20805

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 4366	. . . . 5 ⊢ (𝑥 ⊆ {∅, 𝐴} ↔ ((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})))
2		unieq 4444	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
3		uni0 4465	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
4		0ex 4790	. . . . . . . . . . 11 ⊢ ∅ ∈ V
5	4	prid1 4297	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, 𝐴}
6	3, 5	eqeltri 2697	. . . . . . . . 9 ⊢ ∪ ∅ ∈ {∅, 𝐴}
7	2, 6	syl6eqel 2709	. . . . . . . 8 ⊢ (𝑥 = ∅ → ∪ 𝑥 ∈ {∅, 𝐴})
8	7	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = ∅ → ∪ 𝑥 ∈ {∅, 𝐴}))
9		unieq 4444	. . . . . . . . 9 ⊢ (𝑥 = {∅} → ∪ 𝑥 = ∪ {∅})
10	4	unisn 4451	. . . . . . . . . 10 ⊢ ∪ {∅} = ∅
11	10, 5	eqeltri 2697	. . . . . . . . 9 ⊢ ∪ {∅} ∈ {∅, 𝐴}
12	9, 11	syl6eqel 2709	. . . . . . . 8 ⊢ (𝑥 = {∅} → ∪ 𝑥 ∈ {∅, 𝐴})
13	12	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = {∅} → ∪ 𝑥 ∈ {∅, 𝐴}))
14	8, 13	jaod 395	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → ∪ 𝑥 ∈ {∅, 𝐴}))
15		unieq 4444	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴} → ∪ 𝑥 = ∪ {𝐴})
16		unisng 4452	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
17	15, 16	sylan9eqr 2678	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {𝐴}) → ∪ 𝑥 = 𝐴)
18		prid2g 4296	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {∅, 𝐴})
19	18	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {𝐴}) → 𝐴 ∈ {∅, 𝐴})
20	17, 19	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {𝐴}) → ∪ 𝑥 ∈ {∅, 𝐴})
21	20	ex 450	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = {𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
22		unieq 4444	. . . . . . . . . 10 ⊢ (𝑥 = {∅, 𝐴} → ∪ 𝑥 = ∪ {∅, 𝐴})
23		uniprg 4450	. . . . . . . . . . . 12 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
24	4, 23	mpan 706	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
25		uncom 3757	. . . . . . . . . . . 12 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
26		un0 3967	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ ∅) = 𝐴
27	25, 26	eqtri 2644	. . . . . . . . . . 11 ⊢ (∅ ∪ 𝐴) = 𝐴
28	24, 27	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ∪ {∅, 𝐴} = 𝐴)
29	22, 28	sylan9eqr 2678	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {∅, 𝐴}) → ∪ 𝑥 = 𝐴)
30	18	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {∅, 𝐴}) → 𝐴 ∈ {∅, 𝐴})
31	29, 30	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = {∅, 𝐴}) → ∪ 𝑥 ∈ {∅, 𝐴})
32	31	ex 450	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
33	21, 32	jaod 395	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴}) → ∪ 𝑥 ∈ {∅, 𝐴}))
34	14, 33	jaod 395	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})) → ∪ 𝑥 ∈ {∅, 𝐴}))
35	1, 34	syl5bi 232	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
36	35	alrimiv 1855	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥(𝑥 ⊆ {∅, 𝐴} → ∪ 𝑥 ∈ {∅, 𝐴}))
37		vex 3203	. . . . . 6 ⊢ 𝑥 ∈ V
38	37	elpr 4198	. . . . 5 ⊢ (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
39		vex 3203	. . . . . . . . 9 ⊢ 𝑦 ∈ V
40	39	elpr 4198	. . . . . . . 8 ⊢ (𝑦 ∈ {∅, 𝐴} ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴))
41		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑦 = ∅)
42	41	ineq2d 3814	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = (𝑥 ∩ ∅))
43		in0 3968	. . . . . . . . . . . . 13 ⊢ (𝑥 ∩ ∅) = ∅
44	42, 43	syl6eq 2672	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = ∅)
45	44, 5	syl6eqel 2709	. . . . . . . . . . 11 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
46	45	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
47		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → 𝑦 = ∅)
48	47	ineq2d 3814	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = (𝑥 ∩ ∅))
49	48, 43	syl6eq 2672	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) = ∅)
50	49, 5	syl6eqel 2709	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
51	50	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = 𝐴 ∧ 𝑦 = ∅) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
52		simpl 473	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → 𝑥 = ∅)
53	52	ineq1d 3813	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (∅ ∩ 𝑦))
54		0in 3969	. . . . . . . . . . . . 13 ⊢ (∅ ∩ 𝑦) = ∅
55	53, 54	syl6eq 2672	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = ∅)
56	55, 5	syl6eqel 2709	. . . . . . . . . . 11 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
57	56	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
58		ineq12 3809	. . . . . . . . . . . . . 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 3822	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ 𝐴) = 𝐴
61	59, 60	syl6eq 2672	. . . . . . . . . . . 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 2701	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
64	63	ex 450	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
65	46, 51, 57, 64	ccased 988	. . . . . . . . 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 232	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → (𝑦 ∈ {∅, 𝐴} → (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
68	67	ralrimiv 2965	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
69	68	ex 450	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → ∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
70	38, 69	syl5bi 232	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {∅, 𝐴} → ∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴}))
71	70	ralrimiv 2965	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥 ∩ 𝑦) ∈ {∅, 𝐴})
72		prex 4909	. . . 4 ⊢ {∅, 𝐴} ∈ V
73		istopg 20700	. . . 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 957	. 2 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ Top)
76	28	eqcomd 2628	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = ∪ {∅, 𝐴})
77		istopon 20717	. 2 ⊢ ({∅, 𝐴} ∈ (TopOn‘𝐴) ↔ ({∅, 𝐴} ∈ Top ∧ 𝐴 = ∪ {∅, 𝐴}))
78	75, 76, 77	sylanbrc 698	1 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  TopOnctopon 20715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-topon 20716

This theorem is referenced by:  indistop  20806  indisuni  20807  indistpsx  20814  indistpsALT  20817  indistps2ALT  20818  cnindis  21096  indishmph  21601  indistgp  21904  topdifinf  33197