Theorem intssuni2 4934

Description: Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)

Assertion

Ref	Expression
intssuni2	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵)

Proof of Theorem intssuni2

Step	Hyp	Ref	Expression
1		intssuni 4931	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
2		uniss 4876	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)
3	1, 2	sylan9ssr 3953	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   ≠ wne 2960   ⊆ wss 3907  ∅c0 4288  ∪ cuni 4868  ∩ cint 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-v 3459  df-dif 3910  df-ss 3924  df-nul 4289  df-uni 4869  df-int 4909

This theorem is referenced by:  rintn0  5071  fival  9360  mremre  17646  submre  17647  lssintcl  21054  iundifdifd  32816  iundifdif  32817  bj-ismoored2  37610  bj-ismooredr2  37612  ismrcd1  43291