Theorem pw1ss 4170

Description: Unit power set preserves subset. (Contributed by SF, 3-Feb-2015.)

Assertion

Ref	Expression
pw1ss	⊢ (A ⊆ B → ℘1A ⊆ ℘1B)

Proof of Theorem pw1ss

Step	Hyp	Ref	Expression
1		sspwb 4119	. . 3 ⊢ (A ⊆ B ↔ ℘A ⊆ ℘B)
2		ssrin 3481	. . 3 ⊢ (℘A ⊆ ℘B → (℘A ∩ 1c) ⊆ (℘B ∩ 1c))
3	1, 2	sylbi 187	. 2 ⊢ (A ⊆ B → (℘A ∩ 1c) ⊆ (℘B ∩ 1c))
4		df-pw1 4138	. 2 ⊢ ℘1A = (℘A ∩ 1c)
5		df-pw1 4138	. 2 ⊢ ℘1B = (℘B ∩ 1c)
6	3, 4, 5	3sstr4g 3313	1 ⊢ (A ⊆ B → ℘1A ⊆ ℘1B)

Syntax hints:   → wi 4   ∩ cin 3209   ⊆ wss 3258  ℘cpw 3723  1_cc1c 4135  ℘₁cpw1 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  ax-nin 4079  ax-sn 4088

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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pw1 4138

This theorem is referenced by:  sspw1  4336