Theorem moop2

Description: "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Hypothesis	moop2.1	⊢ 𝐵 ∈ V
Assertion	moop2	⊢ ∃* 𝑥 𝐴 = ⟨ 𝐵 , 𝑥 ⟩

Proof

Step	Hyp	Ref	Expression
1		moop2.1	⊢ 𝐵 ∈ V
2		eqtr2	⊢ ( ( 𝐴 = ⟨ 𝐵 , 𝑥 ⟩ ∧ 𝐴 = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩ ) → ⟨ 𝐵 , 𝑥 ⟩ = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩ )
3		vex	⊢ 𝑥 ∈ V
4	1 3	opth	⊢ ( ⟨ 𝐵 , 𝑥 ⟩ = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩ ↔ ( 𝐵 = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∧ 𝑥 = 𝑦 ) )
5	4	simprbi	⊢ ( ⟨ 𝐵 , 𝑥 ⟩ = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩ → 𝑥 = 𝑦 )
6	2 5	syl	⊢ ( ( 𝐴 = ⟨ 𝐵 , 𝑥 ⟩ ∧ 𝐴 = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩ ) → 𝑥 = 𝑦 )
7	6	gen2	⊢ ∀ 𝑥 ∀ 𝑦 ( ( 𝐴 = ⟨ 𝐵 , 𝑥 ⟩ ∧ 𝐴 = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩ ) → 𝑥 = 𝑦 )
8		nfcsb1v	⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐵
9		nfcv	⊢ Ⅎ 𝑥 𝑦
10	8 9	nfop	⊢ Ⅎ 𝑥 ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩
11	10	nfeq2	⊢ Ⅎ 𝑥 𝐴 = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩
12		csbeq1a	⊢ ( 𝑥 = 𝑦 → 𝐵 = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 )
13		id	⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 )
14	12 13	opeq12d	⊢ ( 𝑥 = 𝑦 → ⟨ 𝐵 , 𝑥 ⟩ = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩ )
15	14	eqeq2d	⊢ ( 𝑥 = 𝑦 → ( 𝐴 = ⟨ 𝐵 , 𝑥 ⟩ ↔ 𝐴 = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩ ) )
16	11 15	mo4f	⊢ ( ∃* 𝑥 𝐴 = ⟨ 𝐵 , 𝑥 ⟩ ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝐴 = ⟨ 𝐵 , 𝑥 ⟩ ∧ 𝐴 = ⟨ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 , 𝑦 ⟩ ) → 𝑥 = 𝑦 ) )
17	7 16	mpbir	⊢ ∃* 𝑥 𝐴 = ⟨ 𝐵 , 𝑥 ⟩