Theorem disjimin 39006

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 38987	1 ⊢ ( Disj 𝑆 → Disj (𝑅 ∩ 𝑆))

Syntax hints:   → wi 4   ∩ cin 3900   Disj wdisjALTV 38413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-coss 38670  df-cnvrefrel 38776  df-funALTV 38937  df-disjALTV 38960

This theorem is referenced by:  disjiminres  39007  eqvreldisj4  39082  eqvrelqseqdisj4  39087