Theorem sspwuni 3739

Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Assertion

Ref	Expression
sspwuni	|- ( A C_ ~P B <-> U. A C_ B )

Proof of Theorem sspwuni

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2560	. . . 4 |- x e. _V
2	1	elpw 3365	. . 3 |- ( x e. ~P B <-> x C_ B )
3	2	ralbii 2330	. 2 |- ( A. x e. A x e. ~P B <-> A. x e. A x C_ B )
4		dfss3 2935	. 2 |- ( A C_ ~P B <-> A. x e. A x e. ~P B )
5		unissb 3610	. 2 |- ( U. A C_ B <-> A. x e. A x C_ B )
6	3, 4, 5	3bitr4i 201	1 |- ( A C_ ~P B <-> U. A C_ B )

Syntax hints:   <-> wb 98   e. wcel 1393   A.wral 2306   C_ wss 2917   ~Pcpw 3359   U.cuni 3580

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-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581

This theorem is referenced by:  pwssb  3740  elpwuni  3741  rintm  3744  dftr4  3859  iotass  4884  tfrlemibfn  5942  unirnioo  8842