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Theorem iunpwss 3743
 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 3699 . . 3 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
2 eliun 3661 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
3 vex 2560 . . . . . 6 𝑦 ∈ V
43elpw 3365 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
54rexbii 2331 . . . 4 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
62, 5bitri 173 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
73elpw 3365 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
8 uniiun 3710 . . . . 5 𝐴 = 𝑥𝐴 𝑥
98sseq2i 2970 . . . 4 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
107, 9bitri 173 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝑥)
111, 6, 103imtr4i 190 . 2 (𝑦 𝑥𝐴 𝒫 𝑥𝑦 ∈ 𝒫 𝐴)
1211ssriv 2949 1 𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  ∃wrex 2307   ⊆ wss 2917  𝒫 cpw 3359  ∪ cuni 3580  ∪ ciun 3657 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581  df-iun 3659 This theorem is referenced by: (None)
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