Theorem intss 4936

Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) (Proof shortened by OpenAI, 25-Mar-2020.)

Assertion

Ref	Expression
intss	⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Proof of Theorem intss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssralv 4018	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
2	1	ss2abdv 4032	. 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} ⊆ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥})
3		dfint2 4915	. 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}
4		dfint2 4915	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
5	2, 3, 4	3sstr4g 4003	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Syntax hints:   → wi 4  {cab 2708  ∀wral 3045   ⊆ wss 3917  ∩ cint 4913

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-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-ral 3046  df-ss 3934  df-int 4914

This theorem is referenced by:  uniintsn  4952  intabs  5307  cofon1  8639  naddssim  8652  fiss  9382  tc2  9702  tcss  9704  tcel  9705  rankval4  9827  cfub  10209  cflm  10210  cflecard  10213  fin23lem26  10285  clsslem  14957  mrcss  17584  lspss  20897  lbsextlem3  21077  aspss  21793  clsss  22948  1stcfb  23339  ufinffr  23823  cofcut1  27835  spanss  31284  fldgenss  33273  ss2mcls  35562  pclssN  39895  dochspss  41379  clss2lem  43607