Theorem intssuni2 4901

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 4898	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
2		uniss 4846	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)
3	1, 2	sylan9ssr 3981	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ≠ wne 3016   ⊆ wss 3936  ∅c0 4291  ∪ cuni 4838  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-uni 4839  df-int 4877

This theorem is referenced by:  rintn0  5030  fival  8876  mremre  16875  submre  16876  lssintcl  19736  iundifdifd  30313  iundifdif  30314  bj-ismoored2  34403  bj-ismooredr2  34405  ismrcd1  39315