Theorem intss 4433

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 3629	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
2	1	ss2abdv 3638	. 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} ⊆ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥})
3		dfint2 4412	. 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}
4		dfint2 4412	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
5	2, 3, 4	3sstr4g 3609	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Syntax hints:   → wi 4  {cab 2596  ∀wral 2896   ⊆ wss 3540  ∩ cint 4410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-in 3547  df-ss 3554  df-int 4411

This theorem is referenced by:  uniintsn  4449  intabs  4752  fiss  8213  tc2  8501  tcss  8503  tcel  8504  rankval4  8613  cfub  8954  cflm  8955  cflecard  8958  fin23lem26  9030  clsslem  13571  mrcss  16099  lspss  18805  lbsextlem3  18981  aspss  19153  clsss  20668  1stcfb  21058  ufinffr  21543  spanss  27591  ss2mcls  30719  pclssN  34198  dochspss  35685  clss2lem  36937