Theorem naddelim

Description: Ordinal less-than is preserved by natural addition. (Contributed by Scott Fenton, 9-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		oveq1	Could not format ( b = A -> ( b +no C ) = ( A +no C ) ) : No typesetting found for |- ( b = A -> ( b +no C ) = ( A +no C ) ) with typecode |-
2	1	eleq1d	Could not format ( b = A -> ( ( b +no C ) e. x <-> ( A +no C ) e. x ) ) : No typesetting found for |- ( b = A -> ( ( b +no C ) e. x <-> ( A +no C ) e. x ) ) with typecode |-
3	2	rspcv	Could not format ( A e. B -> ( A. b e. B ( b +no C ) e. x -> ( A +no C ) e. x ) ) : No typesetting found for |- ( A e. B -> ( A. b e. B ( b +no C ) e. x -> ( A +no C ) e. x ) ) with typecode |-
4	3	ad2antlr	Could not format ( ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) /\ x e. On ) -> ( A. b e. B ( b +no C ) e. x -> ( A +no C ) e. x ) ) : No typesetting found for |- ( ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) /\ x e. On ) -> ( A. b e. B ( b +no C ) e. x -> ( A +no C ) e. x ) ) with typecode |-
5	4	adantld	Could not format ( ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) /\ x e. On ) -> ( ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) -> ( A +no C ) e. x ) ) : No typesetting found for |- ( ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) /\ x e. On ) -> ( ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) -> ( A +no C ) e. x ) ) with typecode |-
6	5	ralrimiva	Could not format ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) -> A. x e. On ( ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) -> ( A +no C ) e. x ) ) : No typesetting found for |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) -> A. x e. On ( ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) -> ( A +no C ) e. x ) ) with typecode |-
7		ovex	Could not format ( A +no C ) e. _V : No typesetting found for |- ( A +no C ) e. _V with typecode |-
8	7	elintrab	Could not format ( ( A +no C ) e. |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } <-> A. x e. On ( ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) -> ( A +no C ) e. x ) ) : No typesetting found for |- ( ( A +no C ) e. |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } <-> A. x e. On ( ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) -> ( A +no C ) e. x ) ) with typecode |-
9	6 8	sylibr	Could not format ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) -> ( A +no C ) e. |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } ) : No typesetting found for |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) -> ( A +no C ) e. |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } ) with typecode |-
10		naddov2	Could not format ( ( B e. On /\ C e. On ) -> ( B +no C ) = |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } ) : No typesetting found for |- ( ( B e. On /\ C e. On ) -> ( B +no C ) = |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } ) with typecode |-
11	10	3adant1	Could not format ( ( A e. On /\ B e. On /\ C e. On ) -> ( B +no C ) = |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } ) : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( B +no C ) = |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } ) with typecode |-
12	11	adantr	Could not format ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) -> ( B +no C ) = |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } ) : No typesetting found for |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) -> ( B +no C ) = |^| { x e. On | ( A. c e. C ( B +no c ) e. x /\ A. b e. B ( b +no C ) e. x ) } ) with typecode |-
13	9 12	eleqtrrd	Could not format ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) -> ( A +no C ) e. ( B +no C ) ) : No typesetting found for |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A e. B ) -> ( A +no C ) e. ( B +no C ) ) with typecode |-
14	13	ex	Could not format ( ( A e. On /\ B e. On /\ C e. On ) -> ( A e. B -> ( A +no C ) e. ( B +no C ) ) ) : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A e. B -> ( A +no C ) e. ( B +no C ) ) ) with typecode |-