Theorem difsymssdifssd 4210

Description: If the symmetric difference is contained in 𝐶, so is one of the differences. (Contributed by AV, 17-Aug-2022.)

Hypothesis

Ref	Expression
difsymssdifssd.1	⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶)

Assertion

Ref	Expression
difsymssdifssd	⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)

Proof of Theorem difsymssdifssd

Step	Hyp	Ref	Expression
1		difsssymdif 4209	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)
2		difsymssdifssd.1	. 2 ⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶)
3	1, 2	sstrid 3942	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∖ cdif 3896   ⊆ wss 3899   △ csymdif 4198

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 3904  df-ss 3916  df-symdif 4199

This theorem is used by:  mbfeqalem1  25901