Theorem intss 4897

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 3983	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
2	1	ss2abdv 3993	. 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} ⊆ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥})
3		dfint2 4878	. 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}
4		dfint2 4878	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
5	2, 3, 4	3sstr4g 3962	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Syntax hints:   → wi 4  {cab 2715  ∀wral 3063   ⊆ wss 3883  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-in 3890  df-ss 3900  df-int 4877

This theorem is referenced by:  uniintsn  4915  intabs  5261  fiss  9113  tc2  9431  tcss  9433  tcel  9434  rankval4  9556  cfub  9936  cflm  9937  cflecard  9940  fin23lem26  10012  clsslem  14623  mrcss  17242  lspss  20161  lbsextlem3  20337  aspss  20991  clsss  22113  1stcfb  22504  ufinffr  22988  spanss  29611  ss2mcls  33430  naddssim  33764  cofcut1  34017  pclssN  37835  dochspss  39319  clss2lem  41108