Theorem difsssymdif 4157

Description: The symmetric difference contains one of the differences. (Proposed by BJ, 18-Aug-2022.) (Contributed by AV, 19-Aug-2022.)

Assertion

Ref	Expression
difsssymdif	⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)

Proof of Theorem difsssymdif

Step	Hyp	Ref	Expression
1		ssun1 4076	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
2		df-symdif 4147	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
3	1, 2	sseqtrri 3928	1 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)

Syntax hints:   ∖ cdif 3854   ∪ cun 3855   ⊆ wss 3857   △ csymdif 4146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2706

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-v 3403  df-un 3862  df-in 3864  df-ss 3874  df-symdif 4147

This theorem is referenced by:  difsymssdifssd  4158