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Theorem iunpwss 4753
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 4697 . . 3 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
2 eliun 4659 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
3 selpw 4305 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
43rexbii 3189 . . . 4 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
52, 4bitri 264 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
6 selpw 4305 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
7 uniiun 4708 . . . . 5 𝐴 = 𝑥𝐴 𝑥
87sseq2i 3779 . . . 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 3756 1 𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  wrex 3062  wss 3723  𝒫 cpw 4298   cuni 4575   ciun 4655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-pw 4300  df-uni 4576  df-iun 4657
This theorem is referenced by: (None)
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