Theorem ssdisj 4413

Description: Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
ssdisj	⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)

Proof of Theorem ssdisj

Step	Hyp	Ref	Expression
1		ssrin 4187	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶))
2		eqimss 3989	. . 3 ⊢ ((𝐵 ∩ 𝐶) = ∅ → (𝐵 ∩ 𝐶) ⊆ ∅)
3	1, 2	sylan9ss 3944	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) ⊆ ∅)
4		ss0 4352	. 2 ⊢ ((𝐴 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) = ∅)
5	3, 4	syl 18	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4280

This theorem is used by:  djudisj  6157  fimacnvdisj  6752  marypha1lem  9409  djuin  9980  ackbij1lem16  10293  ackbij1lem18  10295  fin23lem20  10396  fin23lem30  10401  psdmul  22467  elcls3  23381  neindisj  23415  pthhashvtx  30297  imadifxp  33177  ldgenpisyslem1  34778  chtvalz  35241  diophren  43773