Theorem topdifinfeq 35448

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 4535	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
2		sseqin2 4146	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑥) = 𝑥)
3	1, 2	bitri 274	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∩ 𝑥) = 𝑥)
4		eqeq1 2742	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝑥) = 𝑥 → ((𝐴 ∩ 𝑥) = ∅ ↔ 𝑥 = ∅))
5	3, 4	sylbi 216	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((𝐴 ∩ 𝑥) = ∅ ↔ 𝑥 = ∅))
6		disj3 4384	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝑥) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝑥))
7		eqcom 2745	. . . . . . . 8 ⊢ (𝐴 = (𝐴 ∖ 𝑥) ↔ (𝐴 ∖ 𝑥) = 𝐴)
8	6, 7	bitri 274	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑥) = ∅ ↔ (𝐴 ∖ 𝑥) = 𝐴)
9	5, 8	bitr3di 285	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 → (𝑥 = ∅ ↔ (𝐴 ∖ 𝑥) = 𝐴))
10		eqss 3932	. . . . . . . 8 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥))
11		ssdif0 4294	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
12	11	bicomi 223	. . . . . . . . 9 ⊢ ((𝐴 ∖ 𝑥) = ∅ ↔ 𝐴 ⊆ 𝑥)
13	1, 12	anbi12i 626	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) = ∅) ↔ (𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥))
14	10, 13	bitr4i 277	. . . . . . 7 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) = ∅))
15	14	baib 535	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 → (𝑥 = 𝐴 ↔ (𝐴 ∖ 𝑥) = ∅))
16	9, 15	orbi12d 915	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴 ∖ 𝑥) = 𝐴 ∨ (𝐴 ∖ 𝑥) = ∅)))
17		orcom 866	. . . . 5 ⊢ (((𝐴 ∖ 𝑥) = 𝐴 ∨ (𝐴 ∖ 𝑥) = ∅) ↔ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))
18	16, 17	bitrdi 286	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴)))
19	18	orbi2d 912	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))))
20	19	bicomd 222	. 2 ⊢ (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴)) ↔ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
21	20	rabbiia 3396	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 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  {crab 3067   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532

This theorem is referenced by: (None)