Theorem ss2in 4225

Description: Intersection of subclasses. (Contributed by NM, 5-May-2000.)

Assertion

Ref	Expression
ss2in	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷))

Proof of Theorem ss2in

Step	Hyp	Ref	Expression
1		ssrin 4222	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶))
2		sslin 4223	. 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷))
3	1, 2	sylan9ss 3977	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∩ cin 3930   ⊆ wss 3931

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-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-in 3938  df-ss 3948

This theorem is referenced by:  disjxiun  5121  f1un  6843  undomOLD  9079  strleun  17181  dprdss  20017  dprd2da  20030  ablfac1b  20058  tgcl  22912  innei  23068  hausnei2  23296  bwth  23353  fbssfi  23780  fbunfip  23812  fgcl  23821  blin2  24373  vtxdun  29466  vtxdginducedm1  29528  5oai  31647  mayetes3i  31715  mdsl0  32296  neibastop1  36382  ismblfin  37690  heibor1lem  37838  pl42lem2N  40004  pl42lem3N  40005  ntrk2imkb  44028  ssin0  45046  iscnrm3llem2  48891