Theorem intss 4950

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 4032	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
2	1	ss2abdv 4046	. 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} ⊆ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥})
3		dfint2 4929	. 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}
4		dfint2 4929	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
5	2, 3, 4	3sstr4g 4017	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Syntax hints:   → wi 4  {cab 2714  ∀wral 3052   ⊆ wss 3931  ∩ cint 4927

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-ral 3053  df-ss 3948  df-int 4928

This theorem is referenced by:  uniintsn  4966  intabs  5324  cofon1  8689  naddssim  8702  fiss  9441  tc2  9761  tcss  9763  tcel  9764  rankval4  9886  cfub  10268  cflm  10269  cflecard  10272  fin23lem26  10344  clsslem  15008  mrcss  17633  lspss  20946  lbsextlem3  21126  aspss  21842  clsss  22997  1stcfb  23388  ufinffr  23872  cofcut1  27885  spanss  31334  fldgenss  33315  ss2mcls  35595  pclssN  39918  dochspss  41402  clss2lem  43602