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Theorem iunpwss 4995
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 4936 . . 3 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
2 eliun 4888 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
3 velpw 4505 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
43rexbii 3213 . . . 4 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
52, 4bitri 278 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
6 velpw 4505 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
7 uniiun 4948 . . . . 5 𝐴 = 𝑥𝐴 𝑥
87sseq2i 3947 . . . 4 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
96, 8bitri 278 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝑥)
101, 5, 93imtr4i 295 . 2 (𝑦 𝑥𝐴 𝒫 𝑥𝑦 ∈ 𝒫 𝐴)
1110ssriv 3922 1 𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2112  wrex 3110  wss 3884  𝒫 cpw 4500   cuni 4803   ciun 4884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502  df-uni 4804  df-iun 4886
This theorem is referenced by: (None)
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