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Theorem topdifinfeq 33527
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 4164 . . . . . . . 8 ((𝐴𝑥) = ∅ ↔ 𝐴 = (𝐴𝑥))
2 eqcom 2767 . . . . . . . 8 (𝐴 = (𝐴𝑥) ↔ (𝐴𝑥) = 𝐴)
31, 2bitri 264 . . . . . . 7 ((𝐴𝑥) = ∅ ↔ (𝐴𝑥) = 𝐴)
4 selpw 4309 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
5 sseqin2 3960 . . . . . . . . 9 (𝑥𝐴 ↔ (𝐴𝑥) = 𝑥)
64, 5bitri 264 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴𝑥) = 𝑥)
7 eqeq1 2764 . . . . . . . 8 ((𝐴𝑥) = 𝑥 → ((𝐴𝑥) = ∅ ↔ 𝑥 = ∅))
86, 7sylbi 207 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴 → ((𝐴𝑥) = ∅ ↔ 𝑥 = ∅))
93, 8syl5rbbr 275 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥 = ∅ ↔ (𝐴𝑥) = 𝐴))
10 eqss 3759 . . . . . . . 8 (𝑥 = 𝐴 ↔ (𝑥𝐴𝐴𝑥))
11 ssdif0 4085 . . . . . . . . . 10 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
1211bicomi 214 . . . . . . . . 9 ((𝐴𝑥) = ∅ ↔ 𝐴𝑥)
134, 12anbi12i 735 . . . . . . . 8 ((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅) ↔ (𝑥𝐴𝐴𝑥))
1410, 13bitr4i 267 . . . . . . 7 (𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅))
1514baib 982 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥 = 𝐴 ↔ (𝐴𝑥) = ∅))
169, 15orbi12d 748 . . . . 5 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅)))
17 orcom 401 . . . . 5 (((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))
1816, 17syl6bb 276 . . . 4 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴)))
1918orbi2d 740 . . 3 (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))))
2019bicomd 213 . 2 (𝑥 ∈ 𝒫 𝐴 → ((¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴)) ↔ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2120rabbiia 3324 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 196  wo 382  wa 383   = wceq 1632  wcel 2139  {crab 3054  cdif 3712  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  Fincfn 8123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304
This theorem is referenced by: (None)
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