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Definition df-ltfin 4441
 Description: Define the less than relationship for finite cardinals. Definition from [Rosser] p. 527. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
df-ltfin <fin = {x mn(x = ⟪m, n (m p Nn n = ((m +c p) +c 1c)))}
Distinct variable group:   m,n,p,x

Detailed syntax breakdown of Definition df-ltfin
StepHypRef Expression
1 cltfin 4433 . 2 class <fin
2 vx . . . . . . . 8 setvar x
32cv 1641 . . . . . . 7 class x
4 vm . . . . . . . . 9 setvar m
54cv 1641 . . . . . . . 8 class m
6 vn . . . . . . . . 9 setvar n
76cv 1641 . . . . . . . 8 class n
85, 7copk 4057 . . . . . . 7 class m, n
93, 8wceq 1642 . . . . . 6 wff x = ⟪m, n
10 c0 3550 . . . . . . . 8 class
115, 10wne 2516 . . . . . . 7 wff m
12 vp . . . . . . . . . . . 12 setvar p
1312cv 1641 . . . . . . . . . . 11 class p
145, 13cplc 4375 . . . . . . . . . 10 class (m +c p)
15 c1c 4134 . . . . . . . . . 10 class 1c
1614, 15cplc 4375 . . . . . . . . 9 class ((m +c p) +c 1c)
177, 16wceq 1642 . . . . . . . 8 wff n = ((m +c p) +c 1c)
18 cnnc 4373 . . . . . . . 8 class Nn
1917, 12, 18wrex 2615 . . . . . . 7 wff p Nn n = ((m +c p) +c 1c)
2011, 19wa 358 . . . . . 6 wff (m p Nn n = ((m +c p) +c 1c))
219, 20wa 358 . . . . 5 wff (x = ⟪m, n (m p Nn n = ((m +c p) +c 1c)))
2221, 6wex 1541 . . . 4 wff n(x = ⟪m, n (m p Nn n = ((m +c p) +c 1c)))
2322, 4wex 1541 . . 3 wff mn(x = ⟪m, n (m p Nn n = ((m +c p) +c 1c)))
2423, 2cab 2339 . 2 class {x mn(x = ⟪m, n (m p Nn n = ((m +c p) +c 1c)))}
251, 24wceq 1642 1 wff <fin = {x mn(x = ⟪m, n (m p Nn n = ((m +c p) +c 1c)))}
 Colors of variables: wff setvar class This definition is referenced by:  opkltfing  4449  ltfinex  4464
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