Theorem dju0en

Description: Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of Enderton p. 143. (Contributed by NM, 27-Sep-2004) (Revised by Mario Carneiro, 29-Apr-2015)

	Ref	Expression
Assertion	dju0en	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ⊔ ∅ ) ≈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		0ex	⊢ ∅ ∈ V
2		in0	⊢ ( 𝐴 ∩ ∅ ) = ∅
3		endjudisj	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∅ ∈ V ∧ ( 𝐴 ∩ ∅ ) = ∅ ) → ( 𝐴 ⊔ ∅ ) ≈ ( 𝐴 ∪ ∅ ) )
4	1 2 3	mp3an23	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ⊔ ∅ ) ≈ ( 𝐴 ∪ ∅ ) )
5		un0	⊢ ( 𝐴 ∪ ∅ ) = 𝐴
6	4 5	breqtrdi	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ⊔ ∅ ) ≈ 𝐴 )