Theorem ssdisj 3580

Description: Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)

Assertion

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

Proof of Theorem ssdisj

Step	Hyp	Ref	Expression
1		ss0b 3562	. . . 4 ⊢ ((𝐵 ∩ 𝐶) ⊆ ∅ ↔ (𝐵 ∩ 𝐶) = ∅)
2		ssrin 3456	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶))
3		sstr2 3255	. . . . 5 ⊢ ((𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶) → ((𝐵 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) ⊆ ∅))
4	2, 3	syl 14	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) ⊆ ∅))
5	1, 4	biimtrrid 153	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) = ∅ → (𝐴 ∩ 𝐶) ⊆ ∅))
6	5	imp 124	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) ⊆ ∅)
7		ss0 3563	. 2 ⊢ ((𝐴 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) = ∅)
8	6, 7	syl 14	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by:  djudisj  5210  fimacnvdisj  5571  unfiin  7223  hashunlem  11222