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Theorem moop2 5512
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.
StepHypRef Expression
1 eqtr2 2759 . . . 4 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
2 moop2.1 . . . . . 6 𝐵 ∈ V
3 vex 3482 . . . . . 6 𝑥 ∈ V
42, 3opth 5487 . . . . 5 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ ↔ (𝐵 = 𝑦 / 𝑥𝐵𝑥 = 𝑦))
54simprbi 496 . . . 4 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ → 𝑥 = 𝑦)
61, 5syl 17 . . 3 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
76gen2 1793 . 2 𝑥𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8 nfcsb1v 3933 . . . . 5 𝑥𝑦 / 𝑥𝐵
9 nfcv 2903 . . . . 5 𝑥𝑦
108, 9nfop 4894 . . . 4 𝑥𝑦 / 𝑥𝐵, 𝑦
1110nfeq2 2921 . . 3 𝑥 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦
12 csbeq1a 3922 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
13 id 22 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
1412, 13opeq12d 4886 . . . 4 (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
1514eqeq2d 2746 . . 3 (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩))
1611, 15mo4f 2565 . 2 (∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩ ↔ ∀𝑥𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦))
177, 16mpbir 231 1 ∃*𝑥 𝐴 = ⟨𝐵, 𝑥
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  ∃*wmo 2536  Vcvv 3478  csb 3908  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  euop2  5522
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