Theorem naddss1

Description: Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024)

	Ref	Expression
Assertion	naddss1	Could not format assertion : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B <-> ( A +no C ) C_ ( B +no C ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		naddel1	Could not format ( ( B e. On /\ A e. On /\ C e. On ) -> ( B e. A <-> ( B +no C ) e. ( A +no C ) ) ) : No typesetting found for |- ( ( B e. On /\ A e. On /\ C e. On ) -> ( B e. A <-> ( B +no C ) e. ( A +no C ) ) ) with typecode |-
2	1	3com12	Could not format ( ( A e. On /\ B e. On /\ C e. On ) -> ( B e. A <-> ( B +no C ) e. ( A +no C ) ) ) : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( B e. A <-> ( B +no C ) e. ( A +no C ) ) ) with typecode |-
3	2	notbid	Could not format ( ( A e. On /\ B e. On /\ C e. On ) -> ( -. B e. A <-> -. ( B +no C ) e. ( A +no C ) ) ) : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( -. B e. A <-> -. ( B +no C ) e. ( A +no C ) ) ) with typecode |-
4		ontri1	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left(A\subseteq B\leftrightarrow \neg B\in A\right)$$
5	4	3adant3	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\wedge C\in \mathrm{On}\right)\to \left(A\subseteq B\leftrightarrow \neg B\in A\right)$$
6		naddcl	Could not format ( ( A e. On /\ C e. On ) -> ( A +no C ) e. On ) : No typesetting found for |- ( ( A e. On /\ C e. On ) -> ( A +no C ) e. On ) with typecode |-
7	6	3adant2	Could not format ( ( A e. On /\ B e. On /\ C e. On ) -> ( A +no C ) e. On ) : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A +no C ) e. On ) with typecode |-
8		naddcl	Could not format ( ( B e. On /\ C e. On ) -> ( B +no C ) e. On ) : No typesetting found for |- ( ( B e. On /\ C e. On ) -> ( B +no C ) e. On ) with typecode |-
9	8	3adant1	Could not format ( ( A e. On /\ B e. On /\ C e. On ) -> ( B +no C ) e. On ) : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( B +no C ) e. On ) with typecode |-
10		ontri1	Could not format ( ( ( A +no C ) e. On /\ ( B +no C ) e. On ) -> ( ( A +no C ) C_ ( B +no C ) <-> -. ( B +no C ) e. ( A +no C ) ) ) : No typesetting found for |- ( ( ( A +no C ) e. On /\ ( B +no C ) e. On ) -> ( ( A +no C ) C_ ( B +no C ) <-> -. ( B +no C ) e. ( A +no C ) ) ) with typecode |-
11	7 9 10	syl2anc	Could not format ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A +no C ) C_ ( B +no C ) <-> -. ( B +no C ) e. ( A +no C ) ) ) : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A +no C ) C_ ( B +no C ) <-> -. ( B +no C ) e. ( A +no C ) ) ) with typecode |-
12	3 5 11	3bitr4d	Could not format ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B <-> ( A +no C ) C_ ( B +no C ) ) ) : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B <-> ( A +no C ) C_ ( B +no C ) ) ) with typecode |-