Theorem elpwd 3697

Description: Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
elpwd.1	|- ( ph -> A e. V )
elpwd.2	|- ( ph -> A C_ B )

Assertion

Ref	Expression
elpwd	|- ( ph -> A e. ~P B )

Proof of Theorem elpwd

Step	Hyp	Ref	Expression
1		elpwd.2	. 2 |- ( ph -> A C_ B )
2		elpwd.1	. . 3 |- ( ph -> A e. V )
3		elpwg 3696	. . 3 |- ( A e. V -> ( A e. ~P B <-> A C_ B ) )
4	2, 3	syl 14	. 2 |- ( ph -> ( A e. ~P B <-> A C_ B ) )
5	1, 4	mpbird 167	1 |- ( ph -> A e. ~P B )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   e. wcel 2209   C_ wss 3220   ~Pcpw 3688

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690

This theorem is used by:  indval  9296