Theorem difsymssdifssd 4184

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 4183	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)
2		difsymssdifssd.1	. 2 ⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶)
3	1, 2	sstrid 3928	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∖ cdif 3880   ⊆ wss 3883   △ csymdif 4172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-symdif 4173

This theorem is referenced by:  mbfeqalem1  24710