Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-goeq Structured version   Visualization version   GIF version

Definition df-goeq 32588
Description: Define the Godel-set of equality. Here the arguments 𝑥 = ⟨𝑁, 𝑃 correspond to vN and vP , so (∅=𝑔1o) actually means v0 = v1 , not 0 = 1. Here we use the trick mentioned in ax-ext 2790 to introduce equality as a defined notion in terms of 𝑔. The expression suc (𝑢𝑣) = max (𝑢, 𝑣) + 1 here is a convenient way of getting a dummy variable distinct from 𝑢 and 𝑣. (Contributed by Mario Carneiro, 14-Jul-2013.)
Assertion
Ref Expression
df-goeq =𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ suc (𝑢𝑣) / 𝑤𝑔𝑤((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣)))
Distinct variable group:   𝑣,𝑢,𝑤

Detailed syntax breakdown of Definition df-goeq
StepHypRef Expression
1 cgoq 32581 . 2 class =𝑔
2 vu . . 3 setvar 𝑢
3 vv . . 3 setvar 𝑣
4 com 7569 . . 3 class ω
5 vw . . . 4 setvar 𝑤
62cv 1527 . . . . . 6 class 𝑢
73cv 1527 . . . . . 6 class 𝑣
86, 7cun 3931 . . . . 5 class (𝑢𝑣)
98csuc 6186 . . . 4 class suc (𝑢𝑣)
105cv 1527 . . . . . . 7 class 𝑤
11 cgoe 32477 . . . . . . 7 class 𝑔
1210, 6, 11co 7145 . . . . . 6 class (𝑤𝑔𝑢)
1310, 7, 11co 7145 . . . . . 6 class (𝑤𝑔𝑣)
14 cgob 32580 . . . . . 6 class 𝑔
1512, 13, 14co 7145 . . . . 5 class ((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣))
1615, 10cgol 32479 . . . 4 class 𝑔𝑤((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣))
175, 9, 16csb 3880 . . 3 class suc (𝑢𝑣) / 𝑤𝑔𝑤((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣))
182, 3, 4, 4, 17cmpo 7147 . 2 class (𝑢 ∈ ω, 𝑣 ∈ ω ↦ suc (𝑢𝑣) / 𝑤𝑔𝑤((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣)))
191, 18wceq 1528 1 wff =𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ suc (𝑢𝑣) / 𝑤𝑔𝑤((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣)))
Colors of variables: wff setvar class
This definition is referenced by: (None)
  Copyright terms: Public domain W3C validator