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Theorem moop2 4956
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 2640 . . . 4 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
2 moop2.1 . . . . . 6 𝐵 ∈ V
3 vex 3198 . . . . . 6 𝑥 ∈ V
42, 3opth 4935 . . . . 5 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ ↔ (𝐵 = 𝑦 / 𝑥𝐵𝑥 = 𝑦))
54simprbi 480 . . . 4 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ → 𝑥 = 𝑦)
61, 5syl 17 . . 3 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
76gen2 1721 . 2 𝑥𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8 nfcsb1v 3542 . . . . 5 𝑥𝑦 / 𝑥𝐵
9 nfcv 2762 . . . . 5 𝑥𝑦
108, 9nfop 4409 . . . 4 𝑥𝑦 / 𝑥𝐵, 𝑦
1110nfeq2 2777 . . 3 𝑥 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦
12 csbeq1a 3535 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
13 id 22 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
1412, 13opeq12d 4401 . . . 4 (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
1514eqeq2d 2630 . . 3 (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩))
1611, 15mo4f 2514 . 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 384  wal 1479   = wceq 1481  wcel 1988  ∃*wmo 2469  Vcvv 3195  csb 3526  cop 4174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175
This theorem is referenced by:  euop2  4964
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