Theorem elpwi 3698

Description: Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)

Assertion

Ref	Expression
elpwi	|- ( A e. ~P B -> A C_ B )

Proof of Theorem elpwi

Step	Hyp	Ref	Expression
1		elpwg 3696	. 2 |- ( A e. ~P B -> ( A e. ~P B <-> A C_ B ) )
2	1	ibi 176	1 |- ( A e. ~P B -> A C_ B )

This proof depends on syntax axioms:   -> wi 4   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:  elpwid  3700  elelpwi  3701  elpw2g  4292  eldifpw  4623  iunpw  4626  f1opw2  6296  pw1dc1  7221  fi0  7309  2omap  7319  2omapfi  7321  pw1m  7584  pw1on  7586  indval0  9300  hashfibclem  11298  lspf  14810  cnntr  15417  edgssv2en  16606  pw1map  17191