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Theorem iunpwss 3771
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 3727 . . 3 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
2 eliun 3689 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
3 vex 2577 . . . . . 6 𝑦 ∈ V
43elpw 3393 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
54rexbii 2348 . . . 4 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
62, 5bitri 177 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
73elpw 3393 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
8 uniiun 3738 . . . . 5 𝐴 = 𝑥𝐴 𝑥
98sseq2i 2998 . . . 4 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
107, 9bitri 177 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝑥)
111, 6, 103imtr4i 194 . 2 (𝑦 𝑥𝐴 𝒫 𝑥𝑦 ∈ 𝒫 𝐴)
1211ssriv 2977 1 𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1409  wrex 2324  wss 2945  𝒫 cpw 3387   cuni 3608   ciun 3685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-uni 3609  df-iun 3687
This theorem is referenced by: (None)
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