Theorem moop2 5487

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 2784	. . . 4 ⊢ ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
2		moop2.1	. . . . . 6 ⊢ 𝐵 ∈ V
3		vex 3459	. . . . . 6 ⊢ 𝑥 ∈ V
4	2, 3	opth 5460	. . . . 5 ⊢ (⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩ ↔ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝑥 = 𝑦))
5	4	simprbi 502	. . . 4 ⊢ (⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩ → 𝑥 = 𝑦)
6	1, 5	syl 18	. . 3 ⊢ ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦)
7	6	gen2 1826	. 2 ⊢ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8		nfcsb1v 3878	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥𝑦
10	8, 9	nfop 4855	. . . 4 ⊢ Ⅎ𝑥⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
11	10	nfeq2 2942	. . 3 ⊢ Ⅎ𝑥 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
12		csbeq1a 3868	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
13		id 23	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14	12, 13	opeq12d 4847	. . . 4 ⊢ (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
15	14	eqeq2d 2774	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩))
16	11, 15	mo4f 2595	. 2 ⊢ (∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩ ↔ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦))
17	7, 16	mpbir 234	1 ⊢ ∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∃*wmo 2565  Vcvv 3455  ⦋csb 3854  ⟨cop 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597

This theorem is referenced by:  euop2  5497