Theorem topdifinfeq 33198

Description: Two different ways of defining the collection from Exercise 3 of [Munkres] p. 83. (Contributed by ML, 18-Jul-2020.)

Assertion

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

Distinct variable group:   x, A

Proof of Theorem topdifinfeq

Step	Hyp	Ref	Expression
1		disj3 4021	. . . . . . . 8 |- ( ( A i^i x ) = (/) <-> A = ( A \ x ) )
2		eqcom 2629	. . . . . . . 8 |- ( A = ( A \ x ) <-> ( A \ x ) = A )
3	1, 2	bitri 264	. . . . . . 7 |- ( ( A i^i x ) = (/) <-> ( A \ x ) = A )
4		selpw 4165	. . . . . . . . 9 |- ( x e. ~P A <-> x C_ A )
5		sseqin2 3817	. . . . . . . . 9 |- ( x C_ A <-> ( A i^i x ) = x )
6	4, 5	bitri 264	. . . . . . . 8 |- ( x e. ~P A <-> ( A i^i x ) = x )
7		eqeq1 2626	. . . . . . . 8 |- ( ( A i^i x ) = x -> ( ( A i^i x ) = (/) <-> x = (/) ) )
8	6, 7	sylbi 207	. . . . . . 7 |- ( x e. ~P A -> ( ( A i^i x ) = (/) <-> x = (/) ) )
9	3, 8	syl5rbbr 275	. . . . . 6 |- ( x e. ~P A -> ( x = (/) <-> ( A \ x ) = A ) )
10		eqss 3618	. . . . . . . 8 |- ( x = A <-> ( x C_ A /\ A C_ x ) )
11		ssdif0 3942	. . . . . . . . . 10 |- ( A C_ x <-> ( A \ x ) = (/) )
12	11	bicomi 214	. . . . . . . . 9 |- ( ( A \ x ) = (/) <-> A C_ x )
13	4, 12	anbi12i 733	. . . . . . . 8 |- ( ( x e. ~P A /\ ( A \ x ) = (/) ) <-> ( x C_ A /\ A C_ x ) )
14	10, 13	bitr4i 267	. . . . . . 7 |- ( x = A <-> ( x e. ~P A /\ ( A \ x ) = (/) ) )
15	14	baib 944	. . . . . 6 |- ( x e. ~P A -> ( x = A <-> ( A \ x ) = (/) ) )
16	9, 15	orbi12d 746	. . . . 5 |- ( x e. ~P A -> ( ( x = (/) \/ x = A ) <-> ( ( A \ x ) = A \/ ( A \ x ) = (/) ) ) )
17		orcom 402	. . . . 5 |- ( ( ( A \ x ) = A \/ ( A \ x ) = (/) ) <-> ( ( A \ x ) = (/) \/ ( A \ x ) = A ) )
18	16, 17	syl6bb 276	. . . 4 |- ( x e. ~P A -> ( ( x = (/) \/ x = A ) <-> ( ( A \ x ) = (/) \/ ( A \ x ) = A ) ) )
19	18	orbi2d 738	. . 3 |- ( x e. ~P A -> ( ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) <-> ( -. ( A \ x ) e. Fin \/ ( ( A \ x ) = (/) \/ ( A \ x ) = A ) ) ) )
20	19	bicomd 213	. 2 |- ( x e. ~P A -> ( ( -. ( A \ x ) e. Fin \/ ( ( A \ x ) = (/) \/ ( A \ x ) = A ) ) <-> ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) ) )
21	20	rabbiia 3185	1 |- { x e. ~P A | ( -. ( A \ x ) e. Fin \/ ( ( A \ x ) = (/) \/ ( A \ x ) = A ) ) } = { x e. ~P A | ( -. ( A \ x ) e. Fin \/ ( x = (/) \/ x = A ) ) }

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160

This theorem is referenced by: (None)