Theorem eq0

Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of Suppes p. 22. (Contributed by NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024) Avoid ax-8 , df-clel . (Revised by Gino Giotto, 6-Sep-2024)

	Ref	Expression
Assertion	eq0	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		dfnul4	⊢ ∅ = { 𝑦 ∣ ⊥ }
2	1	eqeq2i	⊢ ( 𝐴 = ∅ ↔ 𝐴 = { 𝑦 ∣ ⊥ } )
3		dfcleq	⊢ ( 𝐴 = { 𝑦 ∣ ⊥ } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑦 ∣ ⊥ } ) )
4		df-clab	⊢ ( 𝑥 ∈ { 𝑦 ∣ ⊥ } ↔ [ 𝑥 / 𝑦 ] ⊥ )
5		sbv	⊢ ( [ 𝑥 / 𝑦 ] ⊥ ↔ ⊥ )
6	4 5	bitri	⊢ ( 𝑥 ∈ { 𝑦 ∣ ⊥ } ↔ ⊥ )
7	6	bibi2i	⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑦 ∣ ⊥ } ) ↔ ( 𝑥 ∈ 𝐴 ↔ ⊥ ) )
8		nbfal	⊢ ( ¬ 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ↔ ⊥ ) )
9	7 8	bitr4i	⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑦 ∣ ⊥ } ) ↔ ¬ 𝑥 ∈ 𝐴 )
10	9	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑦 ∣ ⊥ } ) ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
11	3 10	bitri	⊢ ( 𝐴 = { 𝑦 ∣ ⊥ } ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
12	2 11	bitri	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )