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Theorem pw1ss 4169
 Description: Unit power set preserves subset. (Contributed by SF, 3-Feb-2015.)
Assertion
Ref Expression
pw1ss (A B1A 1B)

Proof of Theorem pw1ss
StepHypRef Expression
1 sspwb 4118 . . 3 (A BA B)
2 ssrin 3480 . . 3 (A B → (A ∩ 1c) (B ∩ 1c))
31, 2sylbi 187 . 2 (A B → (A ∩ 1c) (B ∩ 1c))
4 df-pw1 4137 . 2 1A = (A ∩ 1c)
5 df-pw1 4137 . 2 1B = (B ∩ 1c)
63, 4, 53sstr4g 3312 1 (A B1A 1B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ cin 3208   ⊆ wss 3257  ℘cpw 3722  1cc1c 4134  ℘1cpw1 4135 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-sn 4087 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-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pw1 4137 This theorem is referenced by:  sspw1  4335
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