Theorem ssdif 4144

Description: Difference law for subsets. (Contributed by NM, 28-May-1998.)

Assertion

Ref	Expression
ssdif	⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))

Proof of Theorem ssdif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3977	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 611	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
3		eldif 3961	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶))
4		eldif 3961	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
5	2, 3, 4	3imtr4g 296	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐶) → 𝑥 ∈ (𝐵 ∖ 𝐶)))
6	5	ssrdv 3989	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2108   ∖ cdif 3948   ⊆ wss 3951

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-ss 3968

This theorem is referenced by:  ssdifd  4145  pssnn  9208  php  9247  phpOLD  9259  fin1a2lem13  10452  axcclem  10497  isercolllem3  15703  mvdco  19463  dprdres  20048  dpjidcl  20078  ablfac1eulem  20092  cntzsdrg  20803  lspsnat  21147  lbsextlem2  21161  lbsextlem3  21162  cnsubdrglem  21436  mplmonmul  22054  clsconn  23438  2ndcdisj2  23465  kqdisj  23740  nulmbl2  25571  i1f1  25725  itg11  25726  itg1climres  25749  limcresi  25920  dvreslem  25944  dvres2lem  25945  dvaddbr  25974  dvmulbr  25975  dvmulbrOLD  25976  lhop  26055  elqaa  26364  difres  32613  imadifxp  32614  xrge00  33017  elrspunidl  33456  eulerpartlemmf  34377  eulerpartlemgf  34381  bj-2upln1upl  37025  pibt2  37418  mblfinlem3  37666  mblfinlem4  37667  ismblfin  37668  cnambfre  37675  divrngidl  38035  dvrelog2  42065  dvrelog3  42066  readvrec2  42391  readvrec  42392  dffltz  42644  cantnftermord  43333  omabs2  43345  radcnvrat  44333  fourierdlem62  46183