Theorem en0

Description: The empty set is equinumerous only to itself. Exercise 1 of TakeutiZaring p. 88. (Contributed by NM, 27-May-1998) Avoid ax-pow . (Revised by BTernaryTau, 31-Jul-2024)

	Ref	Expression
Assertion	en0	⊢ ( 𝐴 ≈ ∅ ↔ 𝐴 = ∅ )

Proof

Step	Hyp	Ref	Expression
1		bren	⊢ ( 𝐴 ≈ ∅ ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ ∅ )
2		f1ocnv	⊢ ( 𝑓 : 𝐴 –1-1-onto→ ∅ → ◡ 𝑓 : ∅ –1-1-onto→ 𝐴 )
3		f1o00	⊢ ( ◡ 𝑓 : ∅ –1-1-onto→ 𝐴 ↔ ( ◡ 𝑓 = ∅ ∧ 𝐴 = ∅ ) )
4	3	simprbi	⊢ ( ◡ 𝑓 : ∅ –1-1-onto→ 𝐴 → 𝐴 = ∅ )
5	2 4	syl	⊢ ( 𝑓 : 𝐴 –1-1-onto→ ∅ → 𝐴 = ∅ )
6	5	exlimiv	⊢ ( ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ ∅ → 𝐴 = ∅ )
7	1 6	sylbi	⊢ ( 𝐴 ≈ ∅ → 𝐴 = ∅ )
8		0ex	⊢ ∅ ∈ V
9		f1oeq1	⊢ ( 𝑓 = ∅ → ( 𝑓 : ∅ –1-1-onto→ ∅ ↔ ∅ : ∅ –1-1-onto→ ∅ ) )
10		f1o0	⊢ ∅ : ∅ –1-1-onto→ ∅
11	8 9 10	ceqsexv2d	⊢ ∃ 𝑓 𝑓 : ∅ –1-1-onto→ ∅
12		bren	⊢ ( ∅ ≈ ∅ ↔ ∃ 𝑓 𝑓 : ∅ –1-1-onto→ ∅ )
13	11 12	mpbir	⊢ ∅ ≈ ∅
14		breq1	⊢ ( 𝐴 = ∅ → ( 𝐴 ≈ ∅ ↔ ∅ ≈ ∅ ) )
15	13 14	mpbiri	⊢ ( 𝐴 = ∅ → 𝐴 ≈ ∅ )
16	7 15	impbii	⊢ ( 𝐴 ≈ ∅ ↔ 𝐴 = ∅ )