Metamath Proof Explorer


Definition df-nadd

Description: Define natural ordinal addition. This is a commutative form of addition over the ordinals. (Contributed by Scott Fenton, 26-Aug-2024)

Ref Expression
Assertion df-nadd Could not format assertion : No typesetting found for |- +no = frecs ( { <. x , y >. | ( x e. ( On X. On ) /\ y e. ( On X. On ) /\ ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ ( ( 2nd ` x ) _E ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ x =/= y ) ) } , ( On X. On ) , ( z e. _V , a e. _V |-> |^| { w e. On | ( ( a " ( { ( 1st ` z ) } X. ( 2nd ` z ) ) ) C_ w /\ ( a " ( ( 1st ` z ) X. { ( 2nd ` z ) } ) ) C_ w ) } ) ) with typecode |-

Detailed syntax breakdown

Step Hyp Ref Expression
0 cnadd Could not format +no : No typesetting found for class +no with typecode class
1 vx setvar x
2 vy setvar y
3 1 cv setvar x
4 con0 class On
5 4 4 cxp class On × On
6 3 5 wcel wff x On × On
7 2 cv setvar y
8 7 5 wcel wff y On × On
9 c1st class 1 st
10 3 9 cfv class 1 st x
11 cep class E
12 7 9 cfv class 1 st y
13 10 12 11 wbr wff 1 st x E 1 st y
14 10 12 wceq wff 1 st x = 1 st y
15 13 14 wo wff 1 st x E 1 st y 1 st x = 1 st y
16 c2nd class 2 nd
17 3 16 cfv class 2 nd x
18 7 16 cfv class 2 nd y
19 17 18 11 wbr wff 2 nd x E 2 nd y
20 17 18 wceq wff 2 nd x = 2 nd y
21 19 20 wo wff 2 nd x E 2 nd y 2 nd x = 2 nd y
22 3 7 wne wff x y
23 15 21 22 w3a wff 1 st x E 1 st y 1 st x = 1 st y 2 nd x E 2 nd y 2 nd x = 2 nd y x y
24 6 8 23 w3a wff x On × On y On × On 1 st x E 1 st y 1 st x = 1 st y 2 nd x E 2 nd y 2 nd x = 2 nd y x y
25 24 1 2 copab class x y | x On × On y On × On 1 st x E 1 st y 1 st x = 1 st y 2 nd x E 2 nd y 2 nd x = 2 nd y x y
26 vz setvar z
27 cvv class V
28 va setvar a
29 vw setvar w
30 28 cv setvar a
31 26 cv setvar z
32 31 9 cfv class 1 st z
33 32 csn class 1 st z
34 31 16 cfv class 2 nd z
35 33 34 cxp class 1 st z × 2 nd z
36 30 35 cima class a 1 st z × 2 nd z
37 29 cv setvar w
38 36 37 wss wff a 1 st z × 2 nd z w
39 34 csn class 2 nd z
40 32 39 cxp class 1 st z × 2 nd z
41 30 40 cima class a 1 st z × 2 nd z
42 41 37 wss wff a 1 st z × 2 nd z w
43 38 42 wa wff a 1 st z × 2 nd z w a 1 st z × 2 nd z w
44 43 29 4 crab class w On | a 1 st z × 2 nd z w a 1 st z × 2 nd z w
45 44 cint class w On | a 1 st z × 2 nd z w a 1 st z × 2 nd z w
46 26 28 27 27 45 cmpo class z V , a V w On | a 1 st z × 2 nd z w a 1 st z × 2 nd z w
47 5 25 46 cfrecs class frecs x y | x On × On y On × On 1 st x E 1 st y 1 st x = 1 st y 2 nd x E 2 nd y 2 nd x = 2 nd y x y On × On z V , a V w On | a 1 st z × 2 nd z w a 1 st z × 2 nd z w
48 0 47 wceq Could not format +no = frecs ( { <. x , y >. | ( x e. ( On X. On ) /\ y e. ( On X. On ) /\ ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ ( ( 2nd ` x ) _E ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ x =/= y ) ) } , ( On X. On ) , ( z e. _V , a e. _V |-> |^| { w e. On | ( ( a " ( { ( 1st ` z ) } X. ( 2nd ` z ) ) ) C_ w /\ ( a " ( ( 1st ` z ) X. { ( 2nd ` z ) } ) ) C_ w ) } ) ) : No typesetting found for wff +no = frecs ( { <. x , y >. | ( x e. ( On X. On ) /\ y e. ( On X. On ) /\ ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ ( ( 2nd ` x ) _E ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ x =/= y ) ) } , ( On X. On ) , ( z e. _V , a e. _V |-> |^| { w e. On | ( ( a " ( { ( 1st ` z ) } X. ( 2nd ` z ) ) ) C_ w /\ ( a " ( ( 1st ` z ) X. { ( 2nd ` z ) } ) ) C_ w ) } ) ) with typecode wff