Theorem ssin0 39223

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 3840	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∩ 𝐵))
2	1	3adant1 1079	. . 3 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∩ 𝐵))
3		eqimss 3657	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ⊆ ∅)
4	3	3ad2ant1 1082	. . 3 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐴 ∩ 𝐵) ⊆ ∅)
5	2, 4	sstrd 3613	. 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∅)
6		ss0 3974	. 2 ⊢ ((𝐶 ∩ 𝐷) ⊆ ∅ → (𝐶 ∩ 𝐷) = ∅)
7	5, 6	syl 17	1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) = ∅)

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  sge0resplit  40623