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Theorem moop2 5093
 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 2790 . . . 4 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
2 moop2.1 . . . . . 6 𝐵 ∈ V
3 vex 3352 . . . . . 6 𝑥 ∈ V
42, 3opth 5072 . . . . 5 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ ↔ (𝐵 = 𝑦 / 𝑥𝐵𝑥 = 𝑦))
54simprbi 478 . . . 4 (⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩ → 𝑥 = 𝑦)
61, 5syl 17 . . 3 ((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
76gen2 1870 . 2 𝑥𝑦((𝐴 = ⟨𝐵, 𝑥⟩ ∧ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩) → 𝑥 = 𝑦)
8 nfcsb1v 3696 . . . . 5 𝑥𝑦 / 𝑥𝐵
9 nfcv 2912 . . . . 5 𝑥𝑦
108, 9nfop 4553 . . . 4 𝑥𝑦 / 𝑥𝐵, 𝑦
1110nfeq2 2928 . . 3 𝑥 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦
12 csbeq1a 3689 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
13 id 22 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
1412, 13opeq12d 4545 . . . 4 (𝑥 = 𝑦 → ⟨𝐵, 𝑥⟩ = ⟨𝑦 / 𝑥𝐵, 𝑦⟩)
1514eqeq2d 2780 . . 3 (𝑥 = 𝑦 → (𝐴 = ⟨𝐵, 𝑥⟩ ↔ 𝐴 = ⟨𝑦 / 𝑥𝐵, 𝑦⟩))
1611, 15mo4f 2664 . 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 382  ∀wal 1628   = wceq 1630   ∈ wcel 2144  ∃*wmo 2618  Vcvv 3349  ⦋csb 3680  ⟨cop 4320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321 This theorem is referenced by:  euop2  5105
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