Theorem chnpoadomd

Description: A chain under relation which orders the alphabet cannot have more elements than the alphabet itself. (Contributed by Ender Ting, 20-Jan-2026)

	Ref	Expression
Hypotheses	chnpoadomd.1	|- ( ph -> .< Po A )
	chnpoadomd.2	|- ( ph -> B e. ( .< Chain A ) )
	chnpoadomd.3	|- ( ph -> A e. V )
Assertion	chnpoadomd	|- ( ph -> ( 0 ..^ ( # ` B ) ) ~<_ A )

Proof

Step	Hyp	Ref	Expression
1		chnpoadomd.1	|- ( ph -> .< Po A )
2		chnpoadomd.2	|- ( ph -> B e. ( .< Chain A ) )
3		chnpoadomd.3	|- ( ph -> A e. V )
4	1 2	chnpof1	|- ( ph -> B : ( 0 ..^ ( # ` B ) ) -1-1-> A )
5		f1domg	|- ( A e. V -> ( B : ( 0 ..^ ( # ` B ) ) -1-1-> A -> ( 0 ..^ ( # ` B ) ) ~<_ A ) )
6	3 5	syl	|- ( ph -> ( B : ( 0 ..^ ( # ` B ) ) -1-1-> A -> ( 0 ..^ ( # ` B ) ) ~<_ A ) )
7	4 6	mpd	|- ( ph -> ( 0 ..^ ( # ` B ) ) ~<_ A )