Theorem moop2 5392

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 2842	. . . 4 ⊢ ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
2		moop2.1	. . . . . 6 ⊢ 𝐵 ∈ V
3		vex 3497	. . . . . 6 ⊢ 𝑥 ∈ V
4	2, 3	opth 5368	. . . . 5 ⊢ (⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩ ↔ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝑥 = 𝑦))
5	4	simprbi 499	. . . 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 1797	. 2 ⊢ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8		nfcsb1v 3907	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑥𝑦
10	8, 9	nfop 4819	. . . 4 ⊢ Ⅎ𝑥⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
11	10	nfeq2 2995	. . 3 ⊢ Ⅎ𝑥 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩
12		csbeq1a 3897	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
13		id 22	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14	12, 13	opeq12d 4811	. . . 4 ⊢ (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩)
15	14	eqeq2d 2832	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩))
16	11, 15	mo4f 2651	. 2 ⊢ (∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩ ↔ ∀𝑥∀𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨⦋𝑦 / 𝑥⦌𝐵, 𝑦⟩) → 𝑥 = 𝑦))
17	7, 16	mpbir 233	1 ⊢ ∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535   = wceq 1537   ∈ wcel 2114  ∃*wmo 2620  Vcvv 3494  ⦋csb 3883  ⟨cop 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574

This theorem is referenced by:  euop2  5402