Theorem modom2

Description: Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016)

	Ref	Expression
Assertion	modom2	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 ≼ 1o )

Proof

Step	Hyp	Ref	Expression
1		modom	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ { 𝑥 ∣ 𝑥 ∈ 𝐴 } ≼ 1o )
2		abid2	⊢ { 𝑥 ∣ 𝑥 ∈ 𝐴 } = 𝐴
3	2	breq1i	⊢ ( { 𝑥 ∣ 𝑥 ∈ 𝐴 } ≼ 1o ↔ 𝐴 ≼ 1o )
4	1 3	bitri	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 ≼ 1o )