Theorem ssin0 45503

Description: If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
ssin0	⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) = ∅)

Proof of Theorem ssin0

Step	Hyp	Ref	Expression
1		ss2in 4173	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∩ 𝐵))
2	1	3adant1 1136	. . 3 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∩ 𝐵))
3		eqimss 3973	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ⊆ ∅)
4	3	3ad2ant1 1139	. . 3 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐴 ∩ 𝐵) ⊆ ∅)
5	2, 4	sstrd 3925	. 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∅)
6		ss0 4330	. 2 ⊢ ((𝐶 ∩ 𝐷) ⊆ ∅ → (𝐶 ∩ 𝐷) = ∅)
7	5, 6	syl 17	1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) = ∅)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4262

This theorem is referenced by:  sge0resplit  46849