Theorem sspw 3684

Description: The powerclass preserves inclusion. See sspwb 4334 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 4334 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)

Assertion

Ref	Expression
sspw	|- ( A C_ B -> ~P A C_ ~P B )

Proof of Theorem sspw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3247	. . . 4 |- ( x C_ A -> ( A C_ B -> x C_ B ) )
2	1	com12 30	. . 3 |- ( A C_ B -> ( x C_ A -> x C_ B ) )
3		velpw 3678	. . 3 |- ( x e. ~P A <-> x C_ A )
4		velpw 3678	. . 3 |- ( x e. ~P B <-> x C_ B )
5	2, 3, 4	3imtr4g 205	. 2 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
6	5	ssrdv 3246	1 |- ( A C_ B -> ~P A C_ ~P B )

Syntax hints:   -> wi 4   e. wcel 2205   C_ wss 3213   ~Pcpw 3671

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219  df-ss 3226  df-pw 3673

This theorem is referenced by:  sspwi  3685  sspwd  3686