Theorem intss 4642

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 3799	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
2	1	ss2abdv 3808	. 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} ⊆ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥})
3		dfint2 4621	. 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}
4		dfint2 4621	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
5	2, 3, 4	3sstr4g 3779	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)

Syntax hints:   → wi 4  {cab 2738  ∀wral 3042   ⊆ wss 3707  ∩ cint 4619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-in 3714  df-ss 3721  df-int 4620

This theorem is referenced by:  uniintsn  4658  intabs  4966  fiss  8487  tc2  8783  tcss  8785  tcel  8786  rankval4  8895  cfub  9255  cflm  9256  cflecard  9259  fin23lem26  9331  clsslem  13916  mrcss  16470  lspss  19178  lbsextlem3  19354  aspss  19526  clsss  21052  1stcfb  21442  ufinffr  21926  spanss  28508  ss2mcls  31764  pclssN  35675  dochspss  37161  clss2lem  38412