Theorem ssdisj 3494

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 3477	. . . 4 ⊢ ((𝐵 ∩ 𝐶) ⊆ ∅ ↔ (𝐵 ∩ 𝐶) = ∅)
2		ssrin 3375	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶))
3		sstr2 3177	. . . . 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 3478	. 2 ⊢ ((𝐴 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) = ∅)
8	6, 7	syl 14	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∩ cin 3143   ⊆ wss 3144  ∅c0 3437

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438

This theorem is referenced by:  djudisj  5074  fimacnvdisj  5419  unfiin  6955  hashunlem  10819