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Theorem iunpwss 3850
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 3802 . . 3 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
2 eliun 3764 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
3 vex 2644 . . . . . 6 𝑦 ∈ V
43elpw 3463 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
54rexbii 2401 . . . 4 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
62, 5bitri 183 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
73elpw 3463 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
8 uniiun 3813 . . . . 5 𝐴 = 𝑥𝐴 𝑥
98sseq2i 3074 . . . 4 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
107, 9bitri 183 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝑥)
111, 6, 103imtr4i 200 . 2 (𝑦 𝑥𝐴 𝒫 𝑥𝑦 ∈ 𝒫 𝐴)
1211ssriv 3051 1 𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1448  wrex 2376  wss 3021  𝒫 cpw 3457   cuni 3683   ciun 3760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-uni 3684  df-iun 3762
This theorem is referenced by: (None)
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