ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  moop2 GIF version

Theorem moop2 4141
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 2134 . . . 4 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
2 moop2.1 . . . . . 6 𝐵 ∈ V
3 vex 2661 . . . . . 6 𝑥 ∈ V
42, 3opth 4127 . . . . 5 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ ↔ (𝐵 = 𝑦 / 𝑥𝐵𝑥 = 𝑦))
54simprbi 271 . . . 4 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ → 𝑥 = 𝑦)
61, 5syl 14 . . 3 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
76gen2 1409 . 2 𝑥𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8 nfcsb1v 3003 . . . . 5 𝑥𝑦 / 𝑥𝐵
9 nfcv 2256 . . . . 5 𝑥𝑦
108, 9nfop 3689 . . . 4 𝑥𝑦 / 𝑥𝐵, 𝑦
1110nfeq2 2268 . . 3 𝑥 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦
12 csbeq1a 2981 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
13 id 19 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
1412, 13opeq12d 3681 . . . 4 (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
1514eqeq2d 2127 . . 3 (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩))
1611, 15mo4f 2035 . 2 (∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩ ↔ ∀𝑥𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦))
177, 16mpbir 145 1 ∃*𝑥 𝐴 = ⟨𝐵, 𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312   = wceq 1314  wcel 1463  ∃*wmo 1976  Vcvv 2658  csb 2973  cop 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator