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 e. V -> { (/) , A } e. (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 C_ { (/) , A } <-> ( ( x = (/) \/ x = { (/) } ) \/ ( x = { A } \/ x = { (/) , A } ) ) )
2		unieq 4444	. . . . . . . . 9 |- ( x = (/) -> U. x = U. (/) )
3		uni0 4465	. . . . . . . . . 10 |- U. (/) = (/)
4		0ex 4790	. . . . . . . . . . 11 |- (/) e. _V
5	4	prid1 4297	. . . . . . . . . 10 |- (/) e. { (/) , A }
6	3, 5	eqeltri 2697	. . . . . . . . 9 |- U. (/) e. { (/) , A }
7	2, 6	syl6eqel 2709	. . . . . . . 8 |- ( x = (/) -> U. x e. { (/) , A } )
8	7	a1i 11	. . . . . . 7 |- ( A e. V -> ( x = (/) -> U. x e. { (/) , A } ) )
9		unieq 4444	. . . . . . . . 9 |- ( x = { (/) } -> U. x = U. { (/) } )
10	4	unisn 4451	. . . . . . . . . 10 |- U. { (/) } = (/)
11	10, 5	eqeltri 2697	. . . . . . . . 9 |- U. { (/) } e. { (/) , A }
12	9, 11	syl6eqel 2709	. . . . . . . 8 |- ( x = { (/) } -> U. x e. { (/) , A } )
13	12	a1i 11	. . . . . . 7 |- ( A e. V -> ( x = { (/) } -> U. x e. { (/) , A } ) )
14	8, 13	jaod 395	. . . . . 6 |- ( A e. V -> ( ( x = (/) \/ x = { (/) } ) -> U. x e. { (/) , A } ) )
15		unieq 4444	. . . . . . . . . 10 |- ( x = { A } -> U. x = U. { A } )
16		unisng 4452	. . . . . . . . . 10 |- ( A e. V -> U. { A } = A )
17	15, 16	sylan9eqr 2678	. . . . . . . . 9 |- ( ( A e. V /\ x = { A } ) -> U. x = A )
18		prid2g 4296	. . . . . . . . . 10 |- ( A e. V -> A e. { (/) , A } )
19	18	adantr 481	. . . . . . . . 9 |- ( ( A e. V /\ x = { A } ) -> A e. { (/) , A } )
20	17, 19	eqeltrd 2701	. . . . . . . 8 |- ( ( A e. V /\ x = { A } ) -> U. x e. { (/) , A } )
21	20	ex 450	. . . . . . 7 |- ( A e. V -> ( x = { A } -> U. x e. { (/) , A } ) )
22		unieq 4444	. . . . . . . . . 10 |- ( x = { (/) , A } -> U. x = U. { (/) , A } )
23		uniprg 4450	. . . . . . . . . . . 12 |- ( ( (/) e. _V /\ A e. V ) -> U. { (/) , A } = ( (/) u. A ) )
24	4, 23	mpan 706	. . . . . . . . . . 11 |- ( A e. V -> U. { (/) , A } = ( (/) u. A ) )
25		uncom 3757	. . . . . . . . . . . 12 |- ( (/) u. A ) = ( A u. (/) )
26		un0 3967	. . . . . . . . . . . 12 |- ( A u. (/) ) = A
27	25, 26	eqtri 2644	. . . . . . . . . . 11 |- ( (/) u. A ) = A
28	24, 27	syl6eq 2672	. . . . . . . . . 10 |- ( A e. V -> U. { (/) , A } = A )
29	22, 28	sylan9eqr 2678	. . . . . . . . 9 |- ( ( A e. V /\ x = { (/) , A } ) -> U. x = A )
30	18	adantr 481	. . . . . . . . 9 |- ( ( A e. V /\ x = { (/) , A } ) -> A e. { (/) , A } )
31	29, 30	eqeltrd 2701	. . . . . . . 8 |- ( ( A e. V /\ x = { (/) , A } ) -> U. x e. { (/) , A } )
32	31	ex 450	. . . . . . 7 |- ( A e. V -> ( x = { (/) , A } -> U. x e. { (/) , A } ) )
33	21, 32	jaod 395	. . . . . 6 |- ( A e. V -> ( ( x = { A } \/ x = { (/) , A } ) -> U. x e. { (/) , A } ) )
34	14, 33	jaod 395	. . . . 5 |- ( A e. V -> ( ( ( x = (/) \/ x = { (/) } ) \/ ( x = { A } \/ x = { (/) , A } ) ) -> U. x e. { (/) , A } ) )
35	1, 34	syl5bi 232	. . . 4 |- ( A e. V -> ( x C_ { (/) , A } -> U. x e. { (/) , A } ) )
36	35	alrimiv 1855	. . 3 |- ( A e. V -> A. x ( x C_ { (/) , A } -> U. x e. { (/) , A } ) )
37		vex 3203	. . . . . 6 |- x e. _V
38	37	elpr 4198	. . . . 5 |- ( x e. { (/) , A } <-> ( x = (/) \/ x = A ) )
39		vex 3203	. . . . . . . . 9 |- y e. _V
40	39	elpr 4198	. . . . . . . 8 |- ( y e. { (/) , A } <-> ( y = (/) \/ y = A ) )
41		simpr 477	. . . . . . . . . . . . . 14 |- ( ( x = (/) /\ y = (/) ) -> y = (/) )
42	41	ineq2d 3814	. . . . . . . . . . . . 13 |- ( ( x = (/) /\ y = (/) ) -> ( x i^i y ) = ( x i^i (/) ) )
43		in0 3968	. . . . . . . . . . . . 13 |- ( x i^i (/) ) = (/)
44	42, 43	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( x = (/) /\ y = (/) ) -> ( x i^i y ) = (/) )
45	44, 5	syl6eqel 2709	. . . . . . . . . . 11 |- ( ( x = (/) /\ y = (/) ) -> ( x i^i y ) e. { (/) , A } )
46	45	a1i 11	. . . . . . . . . 10 |- ( A e. V -> ( ( x = (/) /\ y = (/) ) -> ( x i^i y ) e. { (/) , A } ) )
47		simpr 477	. . . . . . . . . . . . . 14 |- ( ( x = A /\ y = (/) ) -> y = (/) )
48	47	ineq2d 3814	. . . . . . . . . . . . 13 |- ( ( x = A /\ y = (/) ) -> ( x i^i y ) = ( x i^i (/) ) )
49	48, 43	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( x = A /\ y = (/) ) -> ( x i^i y ) = (/) )
50	49, 5	syl6eqel 2709	. . . . . . . . . . 11 |- ( ( x = A /\ y = (/) ) -> ( x i^i y ) e. { (/) , A } )
51	50	a1i 11	. . . . . . . . . 10 |- ( A e. V -> ( ( x = A /\ y = (/) ) -> ( x i^i y ) e. { (/) , A } ) )
52		simpl 473	. . . . . . . . . . . . . 14 |- ( ( x = (/) /\ y = A ) -> x = (/) )
53	52	ineq1d 3813	. . . . . . . . . . . . 13 |- ( ( x = (/) /\ y = A ) -> ( x i^i y ) = ( (/) i^i y ) )
54		0in 3969	. . . . . . . . . . . . 13 |- ( (/) i^i y ) = (/)
55	53, 54	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( x = (/) /\ y = A ) -> ( x i^i y ) = (/) )
56	55, 5	syl6eqel 2709	. . . . . . . . . . 11 |- ( ( x = (/) /\ y = A ) -> ( x i^i y ) e. { (/) , A } )
57	56	a1i 11	. . . . . . . . . 10 |- ( A e. V -> ( ( x = (/) /\ y = A ) -> ( x i^i y ) e. { (/) , A } ) )
58		ineq12 3809	. . . . . . . . . . . . . 14 |- ( ( x = A /\ y = A ) -> ( x i^i y ) = ( A i^i A ) )
59	58	adantl 482	. . . . . . . . . . . . 13 |- ( ( A e. V /\ ( x = A /\ y = A ) ) -> ( x i^i y ) = ( A i^i A ) )
60		inidm 3822	. . . . . . . . . . . . 13 |- ( A i^i A ) = A
61	59, 60	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( A e. V /\ ( x = A /\ y = A ) ) -> ( x i^i y ) = A )
62	18	adantr 481	. . . . . . . . . . . 12 |- ( ( A e. V /\ ( x = A /\ y = A ) ) -> A e. { (/) , A } )
63	61, 62	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( A e. V /\ ( x = A /\ y = A ) ) -> ( x i^i y ) e. { (/) , A } )
64	63	ex 450	. . . . . . . . . 10 |- ( A e. V -> ( ( x = A /\ y = A ) -> ( x i^i y ) e. { (/) , A } ) )
65	46, 51, 57, 64	ccased 988	. . . . . . . . 9 |- ( A e. V -> ( ( ( x = (/) \/ x = A ) /\ ( y = (/) \/ y = A ) ) -> ( x i^i y ) e. { (/) , A } ) )
66	65	expdimp 453	. . . . . . . 8 |- ( ( A e. V /\ ( x = (/) \/ x = A ) ) -> ( ( y = (/) \/ y = A ) -> ( x i^i y ) e. { (/) , A } ) )
67	40, 66	syl5bi 232	. . . . . . 7 |- ( ( A e. V /\ ( x = (/) \/ x = A ) ) -> ( y e. { (/) , A } -> ( x i^i y ) e. { (/) , A } ) )
68	67	ralrimiv 2965	. . . . . 6 |- ( ( A e. V /\ ( x = (/) \/ x = A ) ) -> A. y e. { (/) , A } ( x i^i y ) e. { (/) , A } )
69	68	ex 450	. . . . 5 |- ( A e. V -> ( ( x = (/) \/ x = A ) -> A. y e. { (/) , A } ( x i^i y ) e. { (/) , A } ) )
70	38, 69	syl5bi 232	. . . 4 |- ( A e. V -> ( x e. { (/) , A } -> A. y e. { (/) , A } ( x i^i y ) e. { (/) , A } ) )
71	70	ralrimiv 2965	. . 3 |- ( A e. V -> A. x e. { (/) , A } A. y e. { (/) , A } ( x i^i y ) e. { (/) , A } )
72		prex 4909	. . . 4 |- { (/) , A } e. _V
73		istopg 20700	. . . 4 |- ( { (/) , A } e. _V -> ( { (/) , A } e. Top <-> ( A. x ( x C_ { (/) , A } -> U. x e. { (/) , A } ) /\ A. x e. { (/) , A } A. y e. { (/) , A } ( x i^i y ) e. { (/) , A } ) ) )
74	72, 73	mp1i 13	. . 3 |- ( A e. V -> ( { (/) , A } e. Top <-> ( A. x ( x C_ { (/) , A } -> U. x e. { (/) , A } ) /\ A. x e. { (/) , A } A. y e. { (/) , A } ( x i^i y ) e. { (/) , A } ) ) )
75	36, 71, 74	mpbir2and 957	. 2 |- ( A e. V -> { (/) , A } e. Top )
76	28	eqcomd 2628	. 2 |- ( A e. V -> A = U. { (/) , A } )
77		istopon 20717	. 2 |- ( { (/) , A } e. (TopOn ` A ) <-> ( { (/) , A } e. Top /\ A = U. { (/) , A } ) )
78	75, 76, 77	sylanbrc 698	1 |- ( A e. V -> { (/) , A } e. (TopOn ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   U.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