Theorem difsssymdif 4208

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 4123	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
2		df-symdif 4198	. 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
3	1, 2	sseqtrri 3979	1 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)

This proof depends on syntax axioms:   ∖ cdif 3895   ∪ cun 3896   ⊆ wss 3898   △ csymdif 4197

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3903  df-ss 3915  df-symdif 4198

This theorem is used by:  difsymssdifssd  4209