Theorem topdifinfindis 33194

Description: Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets x of A such that the complement of x in A is infinite, or x is the empty set, or x is all of A, is the trivial topology when A is finite. (Contributed by ML, 14-Jul-2020.)

Hypothesis

Ref	Expression
topdifinf.t	|- T = { x e. ~P A | ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) }

Assertion

Ref	Expression
topdifinfindis	|- ( A e. Fin -> T = { (/) , A } )

Distinct variable group:   x, A

Allowed substitution hint:   T( x)

Proof of Theorem topdifinfindis

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ x A e. Fin
2		topdifinf.t	. . 3 |- T = { x e. ~P A | ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) }
3		nfrab1 3122	. . 3 |- F/_ x { x e. ~P A | ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) }
4	2, 3	nfcxfr 2762	. 2 |- F/_ x T
5		nfcv 2764	. 2 |- F/_ x { (/) , A }
6		0elpw 4834	. . . . . 6 |- (/) e. ~P A
7		eleq1a 2696	. . . . . 6 |- ( (/) e. ~P A -> ( x = (/) -> x e. ~P A ) )
8	6, 7	mp1i 13	. . . . 5 |- ( A e. Fin -> ( x = (/) -> x e. ~P A ) )
9		pwidg 4173	. . . . . 6 |- ( A e. Fin -> A e. ~P A )
10		eleq1a 2696	. . . . . 6 |- ( A e. ~P A -> ( x = A -> x e. ~P A ) )
11	9, 10	syl 17	. . . . 5 |- ( A e. Fin -> ( x = A -> x e. ~P A ) )
12	8, 11	jaod 395	. . . 4 |- ( A e. Fin -> ( ( x = (/) \/ x = A ) -> x e. ~P A ) )
13	12	pm4.71rd 667	. . 3 |- ( A e. Fin -> ( ( x = (/) \/ x = A ) <-> ( x e. ~P A /\ ( x = (/) \/ x = A ) ) ) )
14		vex 3203	. . . . 5 |- x e. _V
15	14	elpr 4198	. . . 4 |- ( x e. { (/) , A } <-> ( x = (/) \/ x = A ) )
16	15	a1i 11	. . 3 |- ( A e. Fin -> ( x e. { (/) , A } <-> ( x = (/) \/ x = A ) ) )
17		diffi 8192	. . . . . 6 |- ( A e. Fin -> ( A \ x ) e. Fin )
18		biortn 421	. . . . . 6 |- ( ( A \ x ) e. Fin -> ( ( x = (/) \/ x = A ) <-> ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) ) )
19	17, 18	syl 17	. . . . 5 |- ( A e. Fin -> ( ( x = (/) \/ x = A ) <-> ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) ) )
20	19	anbi2d 740	. . . 4 |- ( A e. Fin -> ( ( x e. ~P A /\ ( x = (/) \/ x = A ) ) <-> ( x e. ~P A /\ ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) ) ) )
21	2	rabeq2i 3197	. . . 4 |- ( x e. T <-> ( x e. ~P A /\ ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) ) )
22	20, 21	syl6rbbr 279	. . 3 |- ( A e. Fin -> ( x e. T <-> ( x e. ~P A /\ ( x = (/) \/ x = A ) ) ) )
23	13, 16, 22	3bitr4rd 301	. 2 |- ( A e. Fin -> ( x e. T <-> x e. { (/) , A } ) )
24	1, 4, 5, 23	eqrd 3622	1 |- ( A e. Fin -> T = { (/) , A } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {cpr 4179   Fincfn 7955

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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-om 7066  df-er 7742  df-en 7956  df-fin 7959

This theorem is referenced by:  topdifinf  33197