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Theorem iunpwss 4055
 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 x A
Distinct variable group:   x,A

Proof of Theorem iunpwss
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssiun 4008 . . 3 (x A y xy x A x)
2 eliun 3973 . . . 4 (y x A xx A y x)
3 vex 2862 . . . . . 6 y V
43elpw 3728 . . . . 5 (y xy x)
54rexbii 2639 . . . 4 (x A y xx A y x)
62, 5bitri 240 . . 3 (y x A xx A y x)
73elpw 3728 . . . 4 (y Ay A)
8 uniiun 4019 . . . . 5 A = x A x
98sseq2i 3296 . . . 4 (y Ay x A x)
107, 9bitri 240 . . 3 (y Ay x A x)
111, 6, 103imtr4i 257 . 2 (y x A xy A)
1211ssriv 3277 1 x A x A
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1710  ∃wrex 2615   ⊆ wss 3257  ℘cpw 3722  ∪cuni 3891  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724  df-uni 3892  df-iun 3971 This theorem is referenced by: (None)
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