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Theorem inunissunidif 34764
Description: Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.)
Assertion
Ref Expression
inunissunidif ((𝐴 𝐶) = ∅ → (𝐴 𝐵𝐴 (𝐵𝐶)))

Proof of Theorem inunissunidif
StepHypRef Expression
1 reldisj 4385 . . . 4 (𝐴 𝐵 → ((𝐴 𝐶) = ∅ ↔ 𝐴 ⊆ ( 𝐵 𝐶)))
2 difunieq 34763 . . . . 5 ( 𝐵 𝐶) ⊆ (𝐵𝐶)
3 sstr 3961 . . . . 5 ((𝐴 ⊆ ( 𝐵 𝐶) ∧ ( 𝐵 𝐶) ⊆ (𝐵𝐶)) → 𝐴 (𝐵𝐶))
42, 3mpan2 690 . . . 4 (𝐴 ⊆ ( 𝐵 𝐶) → 𝐴 (𝐵𝐶))
51, 4syl6bi 256 . . 3 (𝐴 𝐵 → ((𝐴 𝐶) = ∅ → 𝐴 (𝐵𝐶)))
65com12 32 . 2 ((𝐴 𝐶) = ∅ → (𝐴 𝐵𝐴 (𝐵𝐶)))
7 difss 4094 . . . 4 (𝐵𝐶) ⊆ 𝐵
87unissi 4833 . . 3 (𝐵𝐶) ⊆ 𝐵
9 sstr 3961 . . 3 ((𝐴 (𝐵𝐶) ∧ (𝐵𝐶) ⊆ 𝐵) → 𝐴 𝐵)
108, 9mpan2 690 . 2 (𝐴 (𝐵𝐶) → 𝐴 𝐵)
116, 10impbid1 228 1 ((𝐴 𝐶) = ∅ → (𝐴 𝐵𝐴 (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  cdif 3916  cin 3918  wss 3919  c0 4276   cuni 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4277  df-uni 4825
This theorem is referenced by:  pibt2  34806
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