Theorem disjimin 39558

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 4190	. 2 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑆
2	1	disjssi 39539	1 ⊢ ( Disj 𝑆 → Disj (𝑅 ∩ 𝑆))

This proof depends on syntax axioms:   → wi 4   ∩ cin 3905   Disj wdisjALTV 38926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-coss 39208  df-cnvrefrel 39314  df-funALTV 39474  df-disjALTV 39497

This theorem is used by:  disjiminres  39559  eqvreldisj4  39637  eqvrelqseqdisj4  39653