Theorem sspwd 3683

Description: The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.)

Hypothesis

Ref	Expression
sspwd.1	|- ( ph -> A C_ B )

Assertion

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

Proof of Theorem sspwd

Step	Hyp	Ref	Expression
1		sspwd.1	. 2 |- ( ph -> A C_ B )
2		sspw 3681	. 2 |- ( A C_ B -> ~P A C_ ~P B )
3	1, 2	syl 14	1 |- ( ph -> ~P A C_ ~P B )

Syntax hints:   -> wi 4   C_ wss 3210   ~Pcpw 3668

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-pw 3670

This theorem is referenced by: (None)