Theorem pw1ss1c 4159

Description: A unit power class is a subset of 1_c. (Contributed by SF, 22-Jan-2015.)

Assertion

Ref	Expression
pw1ss1c	|- ~P1 A C_ 1c

Proof of Theorem pw1ss1c

Step	Hyp	Ref	Expression
1		df-pw1 4138	. 2 |- ~P1 A = (~PA i^i 1c)
2		inss2 3477	. 2 |- (~PA i^i 1c) C_ 1c
3	1, 2	eqsstri 3302	1 |- ~P1 A C_ 1c

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:  eqpw1  4163  pw111  4171  eqpw1uni  4331  pw1equn  4332  pw1eqadj  4333  sspw1  4336  sspw12  4337  enpw1pw  6076  ce0lenc1  6240  tlenc1c  6241