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Metamath Proof Explorer

Theorem difsssymdif

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

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

Proof

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