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Theorem iunpwss 4650
 Description: Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
Assertion
Ref Expression
iunpwss 𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem iunpwss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssiun 4594 . . 3 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
2 eliun 4556 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
3 selpw 4198 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
43rexbii 3070 . . . 4 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
52, 4bitri 264 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
6 selpw 4198 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
7 uniiun 4605 . . . . 5 𝐴 = 𝑥𝐴 𝑥
87sseq2i 3663 . . . 4 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
96, 8bitri 264 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝑥)
101, 5, 93imtr4i 281 . 2 (𝑦 𝑥𝐴 𝒫 𝑥𝑦 ∈ 𝒫 𝐴)
1110ssriv 3640 1 𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2030  ∃wrex 2942   ⊆ wss 3607  𝒫 cpw 4191  ∪ cuni 4468  ∪ ciun 4552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-uni 4469  df-iun 4554 This theorem is referenced by: (None)
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