MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunpwss Structured version   Visualization version   GIF version

Theorem iunpwss 4541
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 4488 . . 3 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
2 eliun 4450 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
3 selpw 4110 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
43rexbii 3018 . . . 4 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
52, 4bitri 262 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
6 selpw 4110 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
7 uniiun 4499 . . . . 5 𝐴 = 𝑥𝐴 𝑥
87sseq2i 3588 . . . 4 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
96, 8bitri 262 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝑥)
101, 5, 93imtr4i 279 . 2 (𝑦 𝑥𝐴 𝒫 𝑥𝑦 ∈ 𝒫 𝐴)
1110ssriv 3567 1 𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 1975  wrex 2892  wss 3535  𝒫 cpw 4103   cuni 4362   ciun 4445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-v 3170  df-in 3542  df-ss 3549  df-pw 4105  df-uni 4363  df-iun 4447
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator