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Theorem moop2 4995
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 2671 . . . 4 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
2 moop2.1 . . . . . 6 𝐵 ∈ V
3 vex 3234 . . . . . 6 𝑥 ∈ V
42, 3opth 4974 . . . . 5 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ ↔ (𝐵 = 𝑦 / 𝑥𝐵𝑥 = 𝑦))
54simprbi 479 . . . 4 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ → 𝑥 = 𝑦)
61, 5syl 17 . . 3 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
76gen2 1763 . 2 𝑥𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8 nfcsb1v 3582 . . . . 5 𝑥𝑦 / 𝑥𝐵
9 nfcv 2793 . . . . 5 𝑥𝑦
108, 9nfop 4449 . . . 4 𝑥𝑦 / 𝑥𝐵, 𝑦
1110nfeq2 2809 . . 3 𝑥 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦
12 csbeq1a 3575 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
13 id 22 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
1412, 13opeq12d 4441 . . . 4 (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
1514eqeq2d 2661 . . 3 (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩))
1611, 15mo4f 2545 . 2 (∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩ ↔ ∀𝑥𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦))
177, 16mpbir 221 1 ∃*𝑥 𝐴 = ⟨𝐵, 𝑥
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521   = wceq 1523  wcel 2030  ∃*wmo 2499  Vcvv 3231  csb 3566  cop 4216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217
This theorem is referenced by:  euop2  5003
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