Theorem sscond 4126

Description: If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 4123. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
ssdifd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
sscond	⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))

Proof of Theorem sscond

Step	Hyp	Ref	Expression
1		ssdifd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sscon 4123	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))

Syntax hints:   → wi 4   ∖ cdif 3928   ⊆ wss 3931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-dif 3934  df-ss 3948

This theorem is referenced by:  ssdif2d  4128  eldifeldifsn  4792  fin23lem26  10344  isercoll2  15690  fctop  22947  ntrss  22998  iunconnlem  23370  clsconn  23373  regr1lem  23682  blcld  24449  rrxdstprj1  25366  voliunlem1  25508  elrgspnsubrunlem2  33248  elrspunidl  33448  carsgclctunlem2  34356  salexct  46343  meaiininclem  46495  carageniuncllem2  46531  seposep  48880