Metamath Proof Explorer


Theorem k39o

Description: Values of the _K3 function are ordinals less than 9. (Contributed by BTernaryTau, 11-Oct-2026)

Ref Expression
Assertion k39o Could not format assertion : No typesetting found for |- ( _K3 ` A ) e. 9o with typecode |-

Proof

Step Hyp Ref Expression
1 2fveq3 Could not format ( x = A -> ( 2nd ` ( `' _J ` x ) ) = ( 2nd ` ( `' _J ` A ) ) ) : No typesetting found for |- ( x = A -> ( 2nd ` ( `' _J ` x ) ) = ( 2nd ` ( `' _J ` A ) ) ) with typecode |-
2 dfk32 Could not format _K3 = ( x e. On |-> ( 2nd ` ( `' _J ` x ) ) ) : No typesetting found for |- _K3 = ( x e. On |-> ( 2nd ` ( `' _J ` x ) ) ) with typecode |-
3 fvex Could not format ( 2nd ` ( `' _J ` A ) ) e. _V : No typesetting found for |- ( 2nd ` ( `' _J ` A ) ) e. _V with typecode |-
4 1 2 3 fvmpt Could not format ( A e. On -> ( _K3 ` A ) = ( 2nd ` ( `' _J ` A ) ) ) : No typesetting found for |- ( A e. On -> ( _K3 ` A ) = ( 2nd ` ( `' _J ` A ) ) ) with typecode |-
5 jf1o Could not format _J : ( ( On X. On ) X. 9o ) -1-1-onto-> On : No typesetting found for |- _J : ( ( On X. On ) X. 9o ) -1-1-onto-> On with typecode |-
6 f1ocnvdm Could not format ( ( _J : ( ( On X. On ) X. 9o ) -1-1-onto-> On /\ A e. On ) -> ( `' _J ` A ) e. ( ( On X. On ) X. 9o ) ) : No typesetting found for |- ( ( _J : ( ( On X. On ) X. 9o ) -1-1-onto-> On /\ A e. On ) -> ( `' _J ` A ) e. ( ( On X. On ) X. 9o ) ) with typecode |-
7 5 6 mpan Could not format ( A e. On -> ( `' _J ` A ) e. ( ( On X. On ) X. 9o ) ) : No typesetting found for |- ( A e. On -> ( `' _J ` A ) e. ( ( On X. On ) X. 9o ) ) with typecode |-
8 xp2nd Could not format ( ( `' _J ` A ) e. ( ( On X. On ) X. 9o ) -> ( 2nd ` ( `' _J ` A ) ) e. 9o ) : No typesetting found for |- ( ( `' _J ` A ) e. ( ( On X. On ) X. 9o ) -> ( 2nd ` ( `' _J ` A ) ) e. 9o ) with typecode |-
9 7 8 syl Could not format ( A e. On -> ( 2nd ` ( `' _J ` A ) ) e. 9o ) : No typesetting found for |- ( A e. On -> ( 2nd ` ( `' _J ` A ) ) e. 9o ) with typecode |-
10 4 9 eqeltrd Could not format ( A e. On -> ( _K3 ` A ) e. 9o ) : No typesetting found for |- ( A e. On -> ( _K3 ` A ) e. 9o ) with typecode |-
11 2 fvmptndm Could not format ( -. A e. On -> ( _K3 ` A ) = (/) ) : No typesetting found for |- ( -. A e. On -> ( _K3 ` A ) = (/) ) with typecode |-
12 df-9o Could not format 9o = suc 8o : No typesetting found for |- 9o = suc 8o with typecode |-
13 nsuceq0 Could not format suc 8o =/= (/) : No typesetting found for |- suc 8o =/= (/) with typecode |-
14 12 13 eqnetri Could not format 9o =/= (/) : No typesetting found for |- 9o =/= (/) with typecode |-
15 9on Could not format 9o e. On : No typesetting found for |- 9o e. On with typecode |-
16 on0eln0 Could not format ( 9o e. On -> ( (/) e. 9o <-> 9o =/= (/) ) ) : No typesetting found for |- ( 9o e. On -> ( (/) e. 9o <-> 9o =/= (/) ) ) with typecode |-
17 15 16 ax-mp Could not format ( (/) e. 9o <-> 9o =/= (/) ) : No typesetting found for |- ( (/) e. 9o <-> 9o =/= (/) ) with typecode |-
18 14 17 mpbir Could not format (/) e. 9o : No typesetting found for |- (/) e. 9o with typecode |-
19 11 18 eqeltrdi Could not format ( -. A e. On -> ( _K3 ` A ) e. 9o ) : No typesetting found for |- ( -. A e. On -> ( _K3 ` A ) e. 9o ) with typecode |-
20 10 19 pm2.61i Could not format ( _K3 ` A ) e. 9o : No typesetting found for |- ( _K3 ` A ) e. 9o with typecode |-