Theorem ssdisj 3277

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 3256	. . . 4 ⊢ ((𝐵 ∩ 𝐶) ⊆ ∅ ↔ (𝐵 ∩ 𝐶) = ∅)
2		ssrin 3162	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶))
3		sstr2 2952	. . . . 5 ⊢ ((𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶) → ((𝐵 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) ⊆ ∅))
4	2, 3	syl 14	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) ⊆ ∅))
5	1, 4	syl5bir 142	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) = ∅ → (𝐴 ∩ 𝐶) ⊆ ∅))
6	5	imp 115	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) ⊆ ∅)
7		ss0 3257	. 2 ⊢ ((𝐴 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) = ∅)
8	6, 7	syl 14	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∩ cin 2916   ⊆ wss 2917  ∅c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225

This theorem is referenced by:  djudisj  4750  fimacnvdisj  5074