Theorem ssdisjOLD 4027

Description: Obsolete proof of ssdisj 4026 as of 14-Jul-2021. (Contributed by FL, 24-Jan-2007.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem ssdisjOLD

Step	Hyp	Ref	Expression
1		ss0b 3973	. . . 4 ⊢ ((𝐵 ∩ 𝐶) ⊆ ∅ ↔ (𝐵 ∩ 𝐶) = ∅)
2		ssrin 3838	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶))
3		sstr2 3610	. . . . 5 ⊢ ((𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶) → ((𝐵 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) ⊆ ∅))
4	2, 3	syl 17	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) ⊆ ∅))
5	1, 4	syl5bir 233	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) = ∅ → (𝐴 ∩ 𝐶) ⊆ ∅))
6	5	imp 445	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) ⊆ ∅)
7		ss0 3974	. 2 ⊢ ((𝐴 ∩ 𝐶) ⊆ ∅ → (𝐴 ∩ 𝐶) = ∅)
8	6, 7	syl 17	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)

Syntax hints:   → wi 4   ∧ wa 384   = 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-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: (None)