Theorem eldisjssd 38757

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

Hypothesis

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

Assertion

Ref	Expression
eldisjssd	⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴))

Proof of Theorem eldisjssd

Step	Hyp	Ref	Expression
1		eldisjssd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		eldisjss 38755	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3900   ElDisj weldisj 38230

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 2112  ax-9 2120  ax-11 2159  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-coss 38427  df-cnvrefrel 38543  df-funALTV 38699  df-disjALTV 38722  df-eldisj 38724

This theorem is referenced by: (None)