Theorem pw1sspw 4171

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

Assertion

Ref	Expression
pw1sspw	⊢ ℘1A ⊆ ℘A

Proof of Theorem pw1sspw

Step	Hyp	Ref	Expression
1		df-pw1 4137	. 2 ⊢ ℘1A = (℘A ∩ 1c)
2		inss1 3475	. 2 ⊢ (℘A ∩ 1c) ⊆ ℘A
3	1, 2	eqsstri 3301	1 ⊢ ℘1A ⊆ ℘A

Syntax hints:   ∩ cin 3208   ⊆ wss 3257  ℘cpw 3722  1_cc1c 4134  ℘₁cpw1 4135

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw1 4137

This theorem is referenced by:  ltcpw1pwg  6202