Theorem ssdisj 3601

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 3581	. . . 4 ⊢ ((B ∩ C) ⊆ ∅ ↔ (B ∩ C) = ∅)
2		ssrin 3481	. . . . 5 ⊢ (A ⊆ B → (A ∩ C) ⊆ (B ∩ C))
3		sstr2 3280	. . . . 5 ⊢ ((A ∩ C) ⊆ (B ∩ C) → ((B ∩ C) ⊆ ∅ → (A ∩ C) ⊆ ∅))
4	2, 3	syl 15	. . . 4 ⊢ (A ⊆ B → ((B ∩ C) ⊆ ∅ → (A ∩ C) ⊆ ∅))
5	1, 4	syl5bir 209	. . 3 ⊢ (A ⊆ B → ((B ∩ C) = ∅ → (A ∩ C) ⊆ ∅))
6	5	imp 418	. 2 ⊢ ((A ⊆ B ∧ (B ∩ C) = ∅) → (A ∩ C) ⊆ ∅)
7		ss0 3582	. 2 ⊢ ((A ∩ C) ⊆ ∅ → (A ∩ C) = ∅)
8	6, 7	syl 15	1 ⊢ ((A ⊆ B ∧ (B ∩ C) = ∅) → (A ∩ C) = ∅)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  fimacnvdisj  5245