Theorem sscond 4076

Description: If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 4073. (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 4073	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))

Syntax hints:   → wi 4   ∖ cdif 3880   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-ss 3900

This theorem is referenced by:  ssdif2d  4078  eldifeldifsn  4742  fin23lem26  10238  isercoll2  15622  fctop  22987  ntrss  23038  iunconnlem  23410  clsconn  23413  regr1lem  23722  blcld  24488  rrxdstprj1  25394  voliunlem1  25535  elrgspnsubrunlem2  33329  elrspunidl  33511  carsgclctunlem2  34503  salexct  46777  meaiininclem  46929  carageniuncllem2  46965  seposep  49416