Theorem naddword1

Description: Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Assertion	naddword1	Could not format assertion : No typesetting found for |- ( ( A e. On /\ B e. On ) -> A C_ ( A +no B ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		naddrid	Could not format ( A e. On -> ( A +no (/) ) = A ) : No typesetting found for |- ( A e. On -> ( A +no (/) ) = A ) with typecode |-
2	1	adantr	Could not format ( ( A e. On /\ B e. On ) -> ( A +no (/) ) = A ) : No typesetting found for |- ( ( A e. On /\ B e. On ) -> ( A +no (/) ) = A ) with typecode |-
3		0ss	$$⊢\varnothing \subseteq B$$
4		0elon	$$⊢\varnothing \in \mathrm{On}$$
5		naddss2	Could not format ( ( (/) e. On /\ B e. On /\ A e. On ) -> ( (/) C_ B <-> ( A +no (/) ) C_ ( A +no B ) ) ) : No typesetting found for |- ( ( (/) e. On /\ B e. On /\ A e. On ) -> ( (/) C_ B <-> ( A +no (/) ) C_ ( A +no B ) ) ) with typecode |-
6	4 5	mp3an1	Could not format ( ( B e. On /\ A e. On ) -> ( (/) C_ B <-> ( A +no (/) ) C_ ( A +no B ) ) ) : No typesetting found for |- ( ( B e. On /\ A e. On ) -> ( (/) C_ B <-> ( A +no (/) ) C_ ( A +no B ) ) ) with typecode |-
7	6	ancoms	Could not format ( ( A e. On /\ B e. On ) -> ( (/) C_ B <-> ( A +no (/) ) C_ ( A +no B ) ) ) : No typesetting found for |- ( ( A e. On /\ B e. On ) -> ( (/) C_ B <-> ( A +no (/) ) C_ ( A +no B ) ) ) with typecode |-
8	3 7	mpbii	Could not format ( ( A e. On /\ B e. On ) -> ( A +no (/) ) C_ ( A +no B ) ) : No typesetting found for |- ( ( A e. On /\ B e. On ) -> ( A +no (/) ) C_ ( A +no B ) ) with typecode |-
9	2 8	eqsstrrd	Could not format ( ( A e. On /\ B e. On ) -> A C_ ( A +no B ) ) : No typesetting found for |- ( ( A e. On /\ B e. On ) -> A C_ ( A +no B ) ) with typecode |-