Theorem disjssd 36440

Description: Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)

Hypothesis

Ref	Expression
disjssd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
disjssd	⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴))

Proof of Theorem disjssd

Step	Hyp	Ref	Expression
1		disjssd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		disjss 36438	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3860   Disj wdisjALTV 35961

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-coss 36133  df-cnvrefrel 36239  df-funALTV 36389  df-disjALTV 36412

This theorem is referenced by:  disjeq  36441  eldisjss  36445