Definition df-no 27608

Description: Define the class of surreal numbers. The surreal numbers are a proper class of numbers developed by John H. Conway and introduced by Donald Knuth in 1975. They form a proper class into which all ordered fields can be embedded. The approach we take to defining them was first introduced by Hary Gonshor, and is based on the conception of a "sign expansion" of a surreal number. We define the surreals as ordinal-indexed sequences of 1_o and 2_o, analogous to Gonshor's ( − ) and ( + ).

After introducing this definition, we will abstract away from it using axioms that Norman Alling developed in "Foundations of Analysis over Surreal Number Fields." This is done in an effort to be agnostic towards the exact implementation of surreals. (Contributed by Scott Fenton, 9-Jun-2011.)

Assertion

Ref	Expression
df-no	⊢ No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}

Distinct variable group:   𝑓,𝑎

Detailed syntax breakdown of Definition df-no

Step	Hyp	Ref	Expression
1		csur 27605	. 2 class No
2		va	. . . . . 6 setvar 𝑎
3	2	cv 1540	. . . . 5 class 𝑎
4		c1o 8388	. . . . . 6 class 1o
5		c2o 8389	. . . . . 6 class 2o
6	4, 5	cpr 4580	. . . . 5 class {1o, 2o}
7		vf	. . . . . 6 setvar 𝑓
8	7	cv 1540	. . . . 5 class 𝑓
9	3, 6, 8	wf 6486	. . . 4 wff 𝑓:𝑎⟶{1o, 2o}
10		con0 6315	. . . 4 class On
11	9, 2, 10	wrex 3058	. . 3 wff ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}
12	11, 7	cab 2712	. 2 class {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}
13	1, 12	wceq 1541	1 wff No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}

This definition is referenced by:  elno  27611  elnoOLD  27612  sltso  27642  dfno2  43611