Theorem ss2in 2239

Description: Intersection of subclasses.

Assertion

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

Proof of Theorem ss2in

Step	Hyp	Ref	Expression
1		ssrin 2237	. 2 ⊢ (A ⊆ B → (A ∩ C) ⊆ (B ∩ C))
2		sslin 2238	. 2 ⊢ (C ⊆ D → (B ∩ C) ⊆ (B ∩ D))
3	1, 2	sylan9ss 2078	1 ⊢ ((A ⊆ B ⋀ C ⊆ D) → (A ∩ C) ⊆ (B ∩ D))

Syntax hints:   → wi 3   ⋀ wa 223   ∩ cin 2049   ⊆ wss 2050

This theorem is referenced by:  undom 4444  tgclt 7623  innei 7733  opnin 7866  5oa 9601  mdsl0 10232  fgsb 10555  fgsb2 10560

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056

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