Theorem eldisjssi 39011

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 39010	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))
3	1, 2	ax-mp 5	1 ⊢ ( ElDisj 𝐵 → ElDisj 𝐴)

Syntax hints:   → wi 4   ⊆ wss 3902   ElDisj weldisj 38393

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 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-coss 38673  df-cnvrefrel 38779  df-funALTV 38939  df-disjALTV 38962  df-eldisj 38964

This theorem is referenced by: (None)