Theorem oldssmade

Description: The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024)

	Ref	Expression
Assertion	oldssmade	Could not format assertion : No typesetting found for |- ( _Old ` A ) C_ ( _Made ` A ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		elold	Could not format ( A e. On -> ( x e. ( _Old ` A ) <-> E. b e. A x e. ( _Made ` b ) ) ) : No typesetting found for |- ( A e. On -> ( x e. ( _Old ` A ) <-> E. b e. A x e. ( _Made ` b ) ) ) with typecode |-
2		onelss	$$⊢A\in \mathrm{On}\to \left(b\in A\to b\subseteq A\right)$$
3	2	imp	$$⊢\left(A\in \mathrm{On}\wedge b\in A\right)\to b\subseteq A$$
4		madess	Could not format ( ( A e. On /\ b C_ A ) -> ( _Made ` b ) C_ ( _Made ` A ) ) : No typesetting found for |- ( ( A e. On /\ b C_ A ) -> ( _Made ` b ) C_ ( _Made ` A ) ) with typecode |-
5	3 4	syldan	Could not format ( ( A e. On /\ b e. A ) -> ( _Made ` b ) C_ ( _Made ` A ) ) : No typesetting found for |- ( ( A e. On /\ b e. A ) -> ( _Made ` b ) C_ ( _Made ` A ) ) with typecode |-
6	5	sseld	Could not format ( ( A e. On /\ b e. A ) -> ( x e. ( _Made ` b ) -> x e. ( _Made ` A ) ) ) : No typesetting found for |- ( ( A e. On /\ b e. A ) -> ( x e. ( _Made ` b ) -> x e. ( _Made ` A ) ) ) with typecode |-
7	6	rexlimdva	Could not format ( A e. On -> ( E. b e. A x e. ( _Made ` b ) -> x e. ( _Made ` A ) ) ) : No typesetting found for |- ( A e. On -> ( E. b e. A x e. ( _Made ` b ) -> x e. ( _Made ` A ) ) ) with typecode |-
8	1 7	sylbid	Could not format ( A e. On -> ( x e. ( _Old ` A ) -> x e. ( _Made ` A ) ) ) : No typesetting found for |- ( A e. On -> ( x e. ( _Old ` A ) -> x e. ( _Made ` A ) ) ) with typecode |-
9	8	ssrdv	Could not format ( A e. On -> ( _Old ` A ) C_ ( _Made ` A ) ) : No typesetting found for |- ( A e. On -> ( _Old ` A ) C_ ( _Made ` A ) ) with typecode |-
10		oldf	$$⊢\mathrm{Old}:\mathrm{On}⟶𝒫\mathrm{No}$$
11	10	fdmi	$$⊢\mathrm{dom}\mathrm{Old}=\mathrm{On}$$
12	11	eleq2i	$$⊢A\in \mathrm{dom}\mathrm{Old}\leftrightarrow A\in \mathrm{On}$$
13		ndmfv	$$⊢\neg A\in \mathrm{dom}\mathrm{Old}\to \mathrm{Old}\left(A\right)=\varnothing$$
14	12 13	sylnbir	$$⊢\neg A\in \mathrm{On}\to \mathrm{Old}\left(A\right)=\varnothing$$
15		0ss	Could not format (/) C_ ( _Made ` A ) : No typesetting found for |- (/) C_ ( _Made ` A ) with typecode |-
16	14 15	eqsstrdi	Could not format ( -. A e. On -> ( _Old ` A ) C_ ( _Made ` A ) ) : No typesetting found for |- ( -. A e. On -> ( _Old ` A ) C_ ( _Made ` A ) ) with typecode |-
17	9 16	pm2.61i	Could not format ( _Old ` A ) C_ ( _Made ` A ) : No typesetting found for |- ( _Old ` A ) C_ ( _Made ` A ) with typecode |-