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Theorem iunpwss 3734
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 x A 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 3690 . . 3 (x A yxy x A x)
2 eliun 3652 . . . 4 (y x A 𝒫 xx A y 𝒫 x)
3 vex 2554 . . . . . 6 y V
43elpw 3357 . . . . 5 (y 𝒫 xyx)
54rexbii 2325 . . . 4 (x A y 𝒫 xx A yx)
62, 5bitri 173 . . 3 (y x A 𝒫 xx A yx)
73elpw 3357 . . . 4 (y 𝒫 Ay A)
8 uniiun 3701 . . . . 5 A = x A x
98sseq2i 2964 . . . 4 (y Ay x A x)
107, 9bitri 173 . . 3 (y 𝒫 Ay x A x)
111, 6, 103imtr4i 190 . 2 (y x A 𝒫 xy 𝒫 A)
1211ssriv 2943 1 x A 𝒫 x ⊆ 𝒫 A
Colors of variables: wff set class
Syntax hints:   wcel 1390  wrex 2301  wss 2911  𝒫 cpw 3351   cuni 3571   ciun 3648
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-uni 3572  df-iun 3650
This theorem is referenced by: (None)
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