Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  moel Structured version   Visualization version   GIF version

Theorem moel 29680
Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.)
Assertion
Ref Expression
moel (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem moel
StepHypRef Expression
1 ralcom4 3381 . 2 (∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
2 df-ral 3069 . . 3 (∀𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑦(𝑦𝐴𝑥 = 𝑦))
32ralbii 3132 . 2 (∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦 ↔ ∀𝑥𝐴𝑦(𝑦𝐴𝑥 = 𝑦))
4 alcom 2196 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
5 eleq1w 2836 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
65mo4 2669 . . 3 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
7 df-ral 3069 . . . . 5 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
8 impexp 438 . . . . . 6 (((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
98albii 1898 . . . . 5 (∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝑦𝐴𝑥 = 𝑦)))
107, 9bitr4i 268 . . . 4 (∀𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
1110albii 1898 . . 3 (∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦) ↔ ∀𝑦𝑥((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
124, 6, 113bitr4i 293 . 2 (∃*𝑥 𝑥𝐴 ↔ ∀𝑦𝑥𝐴 (𝑦𝐴𝑥 = 𝑦))
131, 3, 123bitr4ri 294 1 (∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383  wal 1632  wcel 2148  ∃*wmo 2622  wral 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-v 3357
This theorem is referenced by:  disjnf  29739
  Copyright terms: Public domain W3C validator