Theorem pw1sspw 4172

Description: A unit power class is a subset of a power class. (Contributed by SF, 10-Mar-2015.)

Assertion

Ref	Expression
pw1sspw	|- ~P1 A C_ ~PA

Proof of Theorem pw1sspw

Step	Hyp	Ref	Expression
1		df-pw1 4138	. 2 |- ~P1 A = (~PA i^i 1c)
2		inss1 3476	. 2 |- (~PA i^i 1c) C_ ~PA
3	1, 2	eqsstri 3302	1 |- ~P1 A C_ ~PA

Syntax hints:   i^i cin 3209   C_ wss 3258  ~Pcpw 3723  1_cc1c 4135  ~P₁ cpw1 4136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw1 4138

This theorem is referenced by:  ltcpw1pwg  6203