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Theorem topdifinfeq 32374
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 3973 . . . . . . . 8 ((𝐴𝑥) = ∅ ↔ 𝐴 = (𝐴𝑥))
2 eqcom 2617 . . . . . . . 8 (𝐴 = (𝐴𝑥) ↔ (𝐴𝑥) = 𝐴)
31, 2bitri 263 . . . . . . 7 ((𝐴𝑥) = ∅ ↔ (𝐴𝑥) = 𝐴)
4 selpw 4115 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
5 sseqin2 3779 . . . . . . . . 9 (𝑥𝐴 ↔ (𝐴𝑥) = 𝑥)
64, 5bitri 263 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴𝑥) = 𝑥)
7 eqeq1 2614 . . . . . . . 8 ((𝐴𝑥) = 𝑥 → ((𝐴𝑥) = ∅ ↔ 𝑥 = ∅))
86, 7sylbi 206 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴 → ((𝐴𝑥) = ∅ ↔ 𝑥 = ∅))
93, 8syl5rbbr 274 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥 = ∅ ↔ (𝐴𝑥) = 𝐴))
10 eqss 3583 . . . . . . . 8 (𝑥 = 𝐴 ↔ (𝑥𝐴𝐴𝑥))
11 ssdif0 3896 . . . . . . . . . 10 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
1211bicomi 213 . . . . . . . . 9 ((𝐴𝑥) = ∅ ↔ 𝐴𝑥)
134, 12anbi12i 729 . . . . . . . 8 ((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅) ↔ (𝑥𝐴𝐴𝑥))
1410, 13bitr4i 266 . . . . . . 7 (𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅))
1514baib 942 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥 = 𝐴 ↔ (𝐴𝑥) = ∅))
169, 15orbi12d 742 . . . . 5 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅)))
17 orcom 401 . . . . 5 (((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))
1816, 17syl6bb 275 . . . 4 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴)))
1918orbi2d 734 . . 3 (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))))
2019bicomd 212 . 2 (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴)) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2120rabbiia 3161 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 195  wo 382  wa 383   = wceq 1475  wcel 1977  {crab 2900  cdif 3537  cin 3539  wss 3540  c0 3874  𝒫 cpw 4108  Fincfn 7841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110
This theorem is referenced by: (None)
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