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Theorem topdifinfeq 35073
 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 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
Distinct variable group:   𝑥,𝐴

Proof of Theorem topdifinfeq
StepHypRef Expression
1 velpw 4502 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
2 sseqin2 4122 . . . . . . . . 9 (𝑥𝐴 ↔ (𝐴𝑥) = 𝑥)
31, 2bitri 278 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴𝑥) = 𝑥)
4 eqeq1 2762 . . . . . . . 8 ((𝐴𝑥) = 𝑥 → ((𝐴𝑥) = ∅ ↔ 𝑥 = ∅))
53, 4sylbi 220 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴 → ((𝐴𝑥) = ∅ ↔ 𝑥 = ∅))
6 disj3 4353 . . . . . . . 8 ((𝐴𝑥) = ∅ ↔ 𝐴 = (𝐴𝑥))
7 eqcom 2765 . . . . . . . 8 (𝐴 = (𝐴𝑥) ↔ (𝐴𝑥) = 𝐴)
86, 7bitri 278 . . . . . . 7 ((𝐴𝑥) = ∅ ↔ (𝐴𝑥) = 𝐴)
95, 8bitr3di 289 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥 = ∅ ↔ (𝐴𝑥) = 𝐴))
10 eqss 3909 . . . . . . . 8 (𝑥 = 𝐴 ↔ (𝑥𝐴𝐴𝑥))
11 ssdif0 4264 . . . . . . . . . 10 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
1211bicomi 227 . . . . . . . . 9 ((𝐴𝑥) = ∅ ↔ 𝐴𝑥)
131, 12anbi12i 629 . . . . . . . 8 ((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅) ↔ (𝑥𝐴𝐴𝑥))
1410, 13bitr4i 281 . . . . . . 7 (𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅))
1514baib 539 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥 = 𝐴 ↔ (𝐴𝑥) = ∅))
169, 15orbi12d 916 . . . . 5 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅)))
17 orcom 867 . . . . 5 (((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))
1816, 17bitrdi 290 . . . 4 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴)))
1918orbi2d 913 . . 3 (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))))
2019bicomd 226 . 2 (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴)) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2120rabbiia 3384 1 {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {crab 3074   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  𝒫 cpw 4497  Fincfn 8532 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rab 3079  df-v 3411  df-dif 3863  df-in 3867  df-ss 3877  df-nul 4228  df-pw 4499 This theorem is referenced by: (None)
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