Theorem pwuni 2757

Description: A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38.

Assertion

Ref	Expression
pwuni	|- A (_ P~U.A

Proof of Theorem pwuni

Step	Hyp	Ref	Expression
1		elssuni 2526	. . 3 |- (x e. A -> x (_ U.A)
2		visset 1813	. . . 4 |- x e. V
3	2	elpw 2404	. . 3 |- (x e. P~U.A <-> x (_ U.A)
4	1, 3	sylibr 200	. 2 |- (x e. A -> x e. P~U.A)
5	4	ssriv 2069	1 |- A (_ P~U.A

Syntax hints:   e. wcel 958   (_ wss 2047  P~cpw 2401  U.cuni 2503

This theorem is referenced by:  sspwuni 2758  uniexb 2907  rankuni 4698  rankc2 4706  rankxplim 4712  shsspwh 9118  cnfilca 10583  cnfilcaOLD 10584

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402  df-uni 2504

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