Definition df-no 27705

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 27702	. 2 class No
2		va	. . . . . 6 setvar 𝑎
3	2	cv 1536	. . . . 5 class 𝑎
4		c1o 8515	. . . . . 6 class 1o
5		c2o 8516	. . . . . 6 class 2o
6	4, 5	cpr 4650	. . . . 5 class {1o, 2o}
7		vf	. . . . . 6 setvar 𝑓
8	7	cv 1536	. . . . 5 class 𝑓
9	3, 6, 8	wf 6569	. . . 4 wff 𝑓:𝑎⟶{1o, 2o}
10		con0 6395	. . . 4 class On
11	9, 2, 10	wrex 3076	. . 3 wff ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}
12	11, 7	cab 2717	. 2 class {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}
13	1, 12	wceq 1537	1 wff No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}

This definition is referenced by:  elno  27708  elnoOLD  27709  sltso  27739  dfno2  43390