Theorem difsymssdifssd 4225

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 4224	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)
2		difsymssdifssd.1	. 2 ⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶)
3	1, 2	sstrid 3956	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∖ cdif 3910   ⊆ wss 3913   △ csymdif 4213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-symdif 4214

This theorem is referenced by:  mbfeqalem1  25771