Theorem difsymssdifssd 4283

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 4282	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)
2		difsymssdifssd.1	. 2 ⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶)
3	1, 2	sstrid 4020	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∖ cdif 3973   ⊆ wss 3976   △ csymdif 4271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-ss 3993  df-symdif 4272

This theorem is referenced by:  mbfeqalem1  25695