Theorem pwpwssunieq 3953

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 3195	. . 3 |- ( U. x = A -> U. x C_ A )
2	1	ss2abi 3213	. 2 |- { x | U. x = A } C_ { x | U. x C_ A }
3		pwpwab 3952	. 2 |- ~P ~P A = { x | U. x C_ A }
4	2, 3	sseqtrri 3176	1 |- { x | U. x = A } C_ ~P ~P A

Syntax hints:   = wceq 1343   {cab 2151   C_ wss 3115   ~Pcpw 3558   U.cuni 3788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-v 2727  df-in 3121  df-ss 3128  df-pw 3560  df-uni 3789

This theorem is referenced by:  toponsspwpwg  12620  dmtopon  12621