Theorem pwpwssunieq 4033

Description: The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
pwpwssunieq	|- { x | U. x = A } C_ ~P ~P A

Distinct variable group:   x, A

Proof of Theorem pwpwssunieq

Step	Hyp	Ref	Expression
1		eqimss 3258	. . 3 |- ( U. x = A -> U. x C_ A )
2	1	ss2abi 3276	. 2 |- { x | U. x = A } C_ { x | U. x C_ A }
3		pwpwab 4032	. 2 |- ~P ~P A = { x | U. x C_ A }
4	2, 3	sseqtrri 3239	1 |- { x | U. x = A } C_ ~P ~P A

Syntax hints:   = wceq 1375   {cab 2195   C_ wss 3177   ~Pcpw 3629   U.cuni 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-v 2781  df-in 3183  df-ss 3190  df-pw 3631  df-uni 3868

This theorem is referenced by:  toponsspwpwg  14661  dmtopon  14662