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Metamath Proof Explorer

Theorem undif2

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif ). Part of proof of Corollary 6K of Enderton p. 144. (Contributed by NM, 19-May-1998)

		Ref	Expression
	Assertion	undif2	⊢ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = ( 𝐴 ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		uncom	⊢ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 )
2		undif1	⊢ ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 ) = ( 𝐵 ∪ 𝐴 )
3		uncom	⊢ ( 𝐵 ∪ 𝐴 ) = ( 𝐴 ∪ 𝐵 )
4	1 2 3	3eqtri	⊢ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = ( 𝐴 ∪ 𝐵 )