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Metamath Proof Explorer

Theorem unssad

Description: If ( A u. B ) is contained in C , so is A . One-way deduction form of unss . Partial converse of unssd . (Contributed by David Moews, 1-May-2017)

		Ref	Expression
	Hypothesis	unssad.1	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ⊆ 𝐶 )
	Assertion	unssad	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		unssad.1	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ⊆ 𝐶 )
2		unss	⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ↔ ( 𝐴 ∪ 𝐵 ) ⊆ 𝐶 )
3	1 2	sylibr	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) )
4	3	simpld	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )