Theorem moop2 5503

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 2757	. . . 4 ⊢ ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
2		moop2.1	. . . . . 6 ⊢ 𝐵 ∈ V
3		vex 3479	. . . . . 6 ⊢ 𝑥 ∈ V
4	2, 3	opth 5477	. . . . 5 ⊢ (⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩ ↔ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝑥 = 𝑦))
5	4	simprbi 498	. . . 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 1799	. 2 ⊢ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8		nfcsb1v 3919	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥𝑦
10	8, 9	nfop 4890	. . . 4 ⊢ Ⅎ𝑥⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
11	10	nfeq2 2921	. . 3 ⊢ Ⅎ𝑥 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
12		csbeq1a 3908	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
13		id 22	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14	12, 13	opeq12d 4882	. . . 4 ⊢ (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
15	14	eqeq2d 2744	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩))
16	11, 15	mo4f 2562	. 2 ⊢ (∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩ ↔ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦))
17	7, 16	mpbir 230	1 ⊢ ∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1540   = wceq 1542   ∈ wcel 2107  ∃*wmo 2533  Vcvv 3475  ⦋csb 3894  ⟨cop 4635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636

This theorem is referenced by:  euop2  5513