Theorem disjimin 36859

Description: Disjointness condition for intersection. (Contributed by Peter Mazsa, 11-Jun-2021.) (Revised by Peter Mazsa, 28-Sep-2021.)

Assertion

Ref	Expression
disjimin	⊢ ( Disj 𝑆 → Disj (𝑅 ∩ 𝑆))

Proof of Theorem disjimin

Step	Hyp	Ref	Expression
1		inss2 4163	. 2 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑆
2	1	disjssi 36843	1 ⊢ ( Disj 𝑆 → Disj (𝑅 ∩ 𝑆))

Syntax hints:   → wi 4   ∩ cin 3886   Disj wdisjALTV 36367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-coss 36537  df-cnvrefrel 36643  df-funALTV 36793  df-disjALTV 36816

This theorem is referenced by:  disjiminres  36860