Definition df-no 27554

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 27551	. 2 class No
2		va	. . . . . 6 setvar 𝑎
3	2	cv 1539	. . . . 5 class 𝑎
4		c1o 8427	. . . . . 6 class 1o
5		c2o 8428	. . . . . 6 class 2o
6	4, 5	cpr 4591	. . . . 5 class {1o, 2o}
7		vf	. . . . . 6 setvar 𝑓
8	7	cv 1539	. . . . 5 class 𝑓
9	3, 6, 8	wf 6507	. . . 4 wff 𝑓:𝑎⟶{1o, 2o}
10		con0 6332	. . . 4 class On
11	9, 2, 10	wrex 3053	. . 3 wff ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}
12	11, 7	cab 2707	. 2 class {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}
13	1, 12	wceq 1540	1 wff No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}

This definition is referenced by:  elno  27557  elnoOLD  27558  sltso  27588  dfno2  43417