Theorem sspwuni 2758

Description: Subclass relationship for power class and union.

Assertion

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

Proof of Theorem sspwuni

Step	Hyp	Ref	Expression
1		uniss 2521	. . 3 |- (A (_ P~B -> U.A (_ U.P~B)
2		unipw 2756	. . 3 |- U.P~B = B
3	1, 2	syl6ss 2107	. 2 |- (A (_ P~B -> U.A (_ B)
4		sspwb 2755	. . 3 |- (U.A (_ B <-> P~U.A (_ P~B)
5		pwuni 2757	. . . 4 |- A (_ P~U.A
6		sstr 2072	. . . 4 |- ((A (_ P~U.A /\ P~U.A (_ P~B) -> A (_ P~B)
7	5, 6	mpan 695	. . 3 |- (P~U.A (_ P~B -> A (_ P~B)
8	4, 7	sylbi 199	. 2 |- (U.A (_ B -> A (_ P~B)
9	3, 8	impbi 157	1 |- (A (_ P~B <-> U.A (_ B)

Syntax hints:   <-> wb 146   (_ wss 2047  P~cpw 2401  U.cuni 2503

This theorem is referenced by:  pwssb 2760  elpwuni 2761  istps2 7607  ump 10459  fgsb2 10580  sfvlim 10605

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504

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