Theorem disjssd 39154

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 39152	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3889   Disj wdisjALTV 38540

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-coss 38822  df-cnvrefrel 38928  df-funALTV 39088  df-disjALTV 39111

This theorem is referenced by:  disjeq  39155  eldisjss  39159