Theorem difsssymdif 4194

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 4110	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
2		df-symdif 4184	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
3	1, 2	sseqtrri 3966	1 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)

Syntax hints:   ∖ cdif 3882   ∪ cun 3883   ⊆ wss 3885   △ csymdif 4183

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-un 3890  df-ss 3902  df-symdif 4184

This theorem is referenced by:  difsymssdifssd  4195