Theorem moop2 4390

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 2257	. . . 4 ⊢ ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
2		moop2.1	. . . . . 6 ⊢ 𝐵 ∈ V
3		vex 2824	. . . . . 6 ⊢ 𝑥 ∈ V
4	2, 3	opth 4375	. . . . 5 ⊢ (⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩ ↔ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝑥 = 𝑦))
5	4	simprbi 275	. . . 4 ⊢ (⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩ → 𝑥 = 𝑦)
6	1, 5	syl 14	. . 3 ⊢ ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦)
7	6	gen2 1503	. 2 ⊢ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8		nfcsb1v 3180	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑥𝑦
10	8, 9	nfop 3918	. . . 4 ⊢ Ⅎ𝑥⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
11	10	nfeq2 2404	. . 3 ⊢ Ⅎ𝑥 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
12		csbeq1a 3156	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
13		id 19	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14	12, 13	opeq12d 3910	. . . 4 ⊢ (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
15	14	eqeq2d 2250	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩))
16	11, 15	mo4f 2147	. 2 ⊢ (∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩ ↔ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦))
17	7, 16	mpbir 146	1 ⊢ ∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402  ∃*wmo 2087   ∈ wcel 2209  Vcvv 2821  ⦋csb 3147  ⟨cop 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717

This theorem is referenced by: (None)