Theorem uni0b

Description: The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004)

	Ref	Expression
Assertion	uni0b	⊢ ( ∪ 𝐴 = ∅ ↔ 𝐴 ⊆ { ∅ } )

Proof

Step	Hyp	Ref	Expression
1		velsn	⊢ ( 𝑥 ∈ { ∅ } ↔ 𝑥 = ∅ )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ { ∅ } ↔ ∀ 𝑥 ∈ 𝐴 𝑥 = ∅ )
3		dfss3	⊢ ( 𝐴 ⊆ { ∅ } ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ { ∅ } )
4		neq0	⊢ ( ¬ ∪ 𝐴 = ∅ ↔ ∃ 𝑦 𝑦 ∈ ∪ 𝐴 )
5		rexcom4	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 ∈ 𝑥 ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 )
6		neq0	⊢ ( ¬ 𝑥 = ∅ ↔ ∃ 𝑦 𝑦 ∈ 𝑥 )
7	6	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 ∈ 𝑥 )
8		eluni2	⊢ ( 𝑦 ∈ ∪ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 )
9	8	exbii	⊢ ( ∃ 𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 )
10	5 7 9	3bitr4ri	⊢ ( ∃ 𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ )
11		rexnal	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝑥 = ∅ )
12	4 10 11	3bitri	⊢ ( ¬ ∪ 𝐴 = ∅ ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝑥 = ∅ )
13	12	con4bii	⊢ ( ∪ 𝐴 = ∅ ↔ ∀ 𝑥 ∈ 𝐴 𝑥 = ∅ )
14	2 3 13	3bitr4ri	⊢ ( ∪ 𝐴 = ∅ ↔ 𝐴 ⊆ { ∅ } )