Theorem intssuni2m 3926

Description: Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)

Assertion

Ref	Expression
intssuni2m	⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem intssuni2m

Step	Hyp	Ref	Expression
1		intssunim 3924	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ ∪ 𝐴)
2		uniss 3888	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)
3	1, 2	sylan9ssr 3218	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1518   ∈ wcel 2180   ⊆ wss 3177  ∪ cuni 3867  ∩ cint 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-in 3183  df-ss 3190  df-uni 3868  df-int 3903

This theorem is referenced by:  rintm  4037  onintonm  4586  fival  7105  fiuni  7113  lssintclm  14313