Theorem topdifinfeq 37849

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 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}

Distinct variable group:   𝑥,𝐴

Proof of Theorem topdifinfeq

Step	Hyp	Ref	Expression
1		velpw 4562	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
2		sseqin2 4177	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑥) = 𝑥)
3	1, 2	bitri 277	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∩ 𝑥) = 𝑥)
4		eqeq1 2768	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝑥) = 𝑥 → ((𝐴 ∩ 𝑥) = ∅ ↔ 𝑥 = ∅))
5	3, 4	sylbi 219	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((𝐴 ∩ 𝑥) = ∅ ↔ 𝑥 = ∅))
6		disj3 4410	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝑥) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝑥))
7		eqcom 2771	. . . . . . . 8 ⊢ (𝐴 = (𝐴 ∖ 𝑥) ↔ (𝐴 ∖ 𝑥) = 𝐴)
8	6, 7	bitri 277	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑥) = ∅ ↔ (𝐴 ∖ 𝑥) = 𝐴)
9	5, 8	bitr3di 288	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 → (𝑥 = ∅ ↔ (𝐴 ∖ 𝑥) = 𝐴))
10		eqss 3953	. . . . . . . 8 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥))
11		ssdif0 4321	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
12	11	bicomi 226	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑥) = ∅ ↔ 𝐴 ⊆ 𝑥)
13	1, 12	anbi12i 637	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) = ∅) ↔ (𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥))
14	10, 13	bitr4i 280	. . . . . . 7 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) = ∅))
15	14	baib 543	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 → (𝑥 = 𝐴 ↔ (𝐴 ∖ 𝑥) = ∅))
16	9, 15	orbi12d 929	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴 ∖ 𝑥) = 𝐴 ∨ (𝐴 ∖ 𝑥) = ∅)))
17		orcom 881	. . . . 5 ⊢ (((𝐴 ∖ 𝑥) = 𝐴 ∨ (𝐴 ∖ 𝑥) = ∅) ↔ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))
18	16, 17	bitrdi 289	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴)))
19	18	orbi2d 926	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))))
20	19	bicomd 225	. 2 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴)) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
21	20	rabbiia 3420	1 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1562   ∈ wcel 2144  {crab 3416   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557  Fincfn 8929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-in 3913  df-ss 3923  df-nul 4288  df-pw 4559

This theorem is referenced by: (None)