Theorem inss 4202

Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)

Assertion

Ref	Expression
inss	⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Proof of Theorem inss

Step	Hyp	Ref	Expression
1		ssinss1 4200	. 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
2		incom 4163	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
3		ssinss1 4200	. . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝐴) ⊆ 𝐶)
4	2, 3	eqsstrid 3974	. 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
5	1, 4	jaoi 858	1 ⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∨ wo 848   ∩ cin 3902   ⊆ wss 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-in 3910  df-ss 3920

This theorem is referenced by:  pmatcoe1fsupp  22657  ppttop  22963  disjorimxrn  39099  iunrelexp0  44058  ntrclsk3  44426  icccncfext  46245