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Theorem topdifinfeq 32830
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 3993 . . . . . . . 8 ((𝐴𝑥) = ∅ ↔ 𝐴 = (𝐴𝑥))
2 eqcom 2628 . . . . . . . 8 (𝐴 = (𝐴𝑥) ↔ (𝐴𝑥) = 𝐴)
31, 2bitri 264 . . . . . . 7 ((𝐴𝑥) = ∅ ↔ (𝐴𝑥) = 𝐴)
4 selpw 4137 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
5 sseqin2 3795 . . . . . . . . 9 (𝑥𝐴 ↔ (𝐴𝑥) = 𝑥)
64, 5bitri 264 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴𝑥) = 𝑥)
7 eqeq1 2625 . . . . . . . 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 3598 . . . . . . . 8 (𝑥 = 𝐴 ↔ (𝑥𝐴𝐴𝑥))
11 ssdif0 3916 . . . . . . . . . 10 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
1211bicomi 214 . . . . . . . . 9 ((𝐴𝑥) = ∅ ↔ 𝐴𝑥)
134, 12anbi12i 732 . . . . . . . 8 ((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅) ↔ (𝑥𝐴𝐴𝑥))
1410, 13bitr4i 267 . . . . . . 7 (𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) = ∅))
1514baib 943 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥 = 𝐴 ↔ (𝐴𝑥) = ∅))
169, 15orbi12d 745 . . . . 5 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅)))
17 orcom 402 . . . . 5 (((𝐴𝑥) = 𝐴 ∨ (𝐴𝑥) = ∅) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))
1816, 17syl6bb 276 . . . 4 (𝑥 ∈ 𝒫 𝐴 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴)))
1918orbi2d 737 . . 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 3173 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 383  wa 384   = wceq 1480  wcel 1987  {crab 2911  cdif 3552  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  Fincfn 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132
This theorem is referenced by: (None)
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