Theorem moop2 4891

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

Hypothesis

Ref	Expression
moop2.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
moop2	⊢ ∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem moop2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr2 2630	. . . 4 ⊢ ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
2		moop2.1	. . . . . 6 ⊢ 𝐵 ∈ V
3		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
4	2, 3	opth 4871	. . . . 5 ⊢ (⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩ ↔ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝑥 = 𝑦))
5	4	simprbi 479	. . . 4 ⊢ (⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩ → 𝑥 = 𝑦)
6	1, 5	syl 17	. . 3 ⊢ ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦)
7	6	gen2 1714	. 2 ⊢ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8		nfcsb1v 3515	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑥𝑦
10	8, 9	nfop 4356	. . . 4 ⊢ Ⅎ𝑥⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
11	10	nfeq2 2766	. . 3 ⊢ Ⅎ𝑥 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
12		csbeq1a 3508	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
13		id 22	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14	12, 13	opeq12d 4348	. . . 4 ⊢ (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
15	14	eqeq2d 2620	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩))
16	11, 15	mo4f 2504	. 2 ⊢ (∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩ ↔ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦))
17	7, 16	mpbir 220	1 ⊢ ∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  Vcvv 3173  ⦋csb 3499  ⟨cop 4131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132

This theorem is referenced by:  euop2  4899