Theorem eldisjssi 39529

Description: Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)

Hypothesis

Ref	Expression
eldisjssi.1	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
eldisjssi	⊢ ( ElDisj 𝐵 → ElDisj 𝐴)

Proof of Theorem eldisjssi

Step	Hyp	Ref	Expression
1		eldisjssi.1	. 2 ⊢ 𝐴 ⊆ 𝐵
2		eldisjss 39528	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))
3	1, 2	ax-mp 5	1 ⊢ ( ElDisj 𝐵 → ElDisj 𝐴)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3908   ElDisj weldisj 38911

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 2738  ax-sep 5262  ax-pr 5409

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 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-coss 39191  df-cnvrefrel 39297  df-funALTV 39457  df-disjALTV 39480  df-eldisj 39482

This theorem is used by: (None)