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Theorem topdifinfeq 33563
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 disj3 4182 . . . . . . . 8 ((𝐴𝑥) = ∅ ↔ 𝐴 = (𝐴𝑥))
2 eqcom 2772 . . . . . . . 8 (𝐴 = (𝐴𝑥) ↔ (𝐴𝑥) = 𝐴)
31, 2bitri 266 . . . . . . 7 ((𝐴𝑥) = ∅ ↔ (𝐴𝑥) = 𝐴)
4 selpw 4322 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
5 sseqin2 3979 . . . . . . . . 9 (𝑥𝐴 ↔ (𝐴𝑥) = 𝑥)
64, 5bitri 266 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴𝑥) = 𝑥)
7 eqeq1 2769 . . . . . . . 8 ((𝐴𝑥) = 𝑥 → ((𝐴𝑥) = ∅ ↔ 𝑥 = ∅))
86, 7sylbi 208 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴 → ((𝐴𝑥) = ∅ ↔ 𝑥 = ∅))
93, 8syl5rbbr 277 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥 = ∅ ↔ (𝐴𝑥) = 𝐴))
10 eqss 3776 . . . . . . . 8 (𝑥 = 𝐴 ↔ (𝑥𝐴𝐴𝑥))
11 ssdif0 4106 . . . . . . . . . 10 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
1211bicomi 215 . . . . . . . . 9 ((𝐴𝑥) = ∅ ↔ 𝐴𝑥)
134, 12anbi12i 620 . . . . . . . 8 ((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅) ↔ (𝑥𝐴𝐴𝑥))
1410, 13bitr4i 269 . . . . . . 7 (𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅))
1514baib 531 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥 = 𝐴 ↔ (𝐴𝑥) = ∅))
169, 15orbi12d 942 . . . . 5 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅)))
17 orcom 896 . . . . 5 (((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))
1816, 17syl6bb 278 . . . 4 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴)))
1918orbi2d 939 . . 3 (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))))
2019bicomd 214 . 2 (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴)) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2120rabbiia 3333 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 197  wa 384  wo 873   = wceq 1652  wcel 2155  {crab 3059  cdif 3729  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  Fincfn 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rab 3064  df-v 3352  df-dif 3735  df-in 3739  df-ss 3746  df-nul 4080  df-pw 4317
This theorem is referenced by: (None)
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