Theorem iunpwss 4420

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

Distinct variable group:   ,

Proof of Theorem iunpwss

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssiun 4372	. . 3
2		eliun 4335	. . . 4
3		selpw 4019	. . . . 5
4	3	rexbii 2959	. . . 4
5	2, 4	bitri 249	. . 3
6		selpw 4019	. . . 4
7		uniiun 4383	. . . . 5
8	7	sseq2i 3528	. . . 4
9	6, 8	bitri 249	. . 3
10	1, 5, 9	3imtr4i 266	. 2
11	10	ssriv 3507	1

Syntax hints:  e.wcel 1818  E.wrex 2808  C_wss 3475  ~Pcpw 4012  U.cuni 4249  U_ciun 4330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-pw 4014  df-uni 4250  df-iun 4332