Theorem difsymssdifssd 4227

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 4226	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)
2		difsymssdifssd.1	. 2 ⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶)
3	1, 2	sstrid 3975	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∖ cdif 3930   ⊆ wss 3933   △ csymdif 4215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-v 3495  df-un 3938  df-in 3940  df-ss 3949  df-symdif 4216

This theorem is referenced by:  mbfeqalem1  24238