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Theorem ssdisjOLD 4171
 Description: Obsolete proof of ssdisj 4170 as of 14-Jul-2021. (Contributed by FL, 24-Jan-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ssdisjOLD ((𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)

Proof of Theorem ssdisjOLD
StepHypRef Expression
1 ss0b 4116 . . . 4 ((𝐵𝐶) ⊆ ∅ ↔ (𝐵𝐶) = ∅)
2 ssrin 3981 . . . . 5 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
3 sstr2 3751 . . . . 5 ((𝐴𝐶) ⊆ (𝐵𝐶) → ((𝐵𝐶) ⊆ ∅ → (𝐴𝐶) ⊆ ∅))
42, 3syl 17 . . . 4 (𝐴𝐵 → ((𝐵𝐶) ⊆ ∅ → (𝐴𝐶) ⊆ ∅))
51, 4syl5bir 233 . . 3 (𝐴𝐵 → ((𝐵𝐶) = ∅ → (𝐴𝐶) ⊆ ∅))
65imp 444 . 2 ((𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) ⊆ ∅)
7 ss0 4117 . 2 ((𝐴𝐶) ⊆ ∅ → (𝐴𝐶) = ∅)
86, 7syl 17 1 ((𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∩ cin 3714   ⊆ wss 3715  ∅c0 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059 This theorem is referenced by: (None)
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