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Theorem om2noseqlt 28323

Description: Surreal less-than relation for 𝐺. (Contributed by Scott Fenton, 18-Apr-2025.)

**Hypotheses**
Ref	Expression
om2noseq.1	⊢ (𝜑 → 𝐶 ∈ No )
om2noseq.2	⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +_s 1_s )), 𝐶) ↾ ω))
om2noseq.3	⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +_s 1_s )), 𝐶) “ ω))

**Assertion**
Ref	Expression
om2noseqlt	⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) <s (𝐺‘𝐵)))

Distinct variable group: 𝑥,𝐶 Allowed substitution hints: 𝜑(𝑥) 𝐴(𝑥) 𝐵(𝑥) 𝐺(𝑥) 𝑍(𝑥)

Proof of Theorem om2noseqlt Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnaordex2 8695	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑦 ∈ ω (𝐴 +_o suc 𝑦) = 𝐵))
2	1	adantl 481	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 ↔ ∃𝑦 ∈ ω (𝐴 +_o suc 𝑦) = 𝐵))
3		suceq 6461	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
4		df-1o 8522	. . . . . . . . . . 11 ⊢ 1_o = suc ∅
5	3, 4	eqtr4di 2798	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → suc 𝑦 = 1_o)
6	5	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝐴 +_o suc 𝑦) = (𝐴 +_o 1_o))
7	6	fveq2d 6924	. . . . . . . 8 ⊢ (𝑦 = ∅ → (𝐺‘(𝐴 +_o suc 𝑦)) = (𝐺‘(𝐴 +_o 1_o)))
8	7	breq2d 5178	. . . . . . 7 ⊢ (𝑦 = ∅ → ((𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑦)) ↔ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o 1_o))))
9		suceq 6461	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
10	9	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝐴 +_o suc 𝑦) = (𝐴 +_o suc 𝑧))
11	10	fveq2d 6924	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝐺‘(𝐴 +_o suc 𝑦)) = (𝐺‘(𝐴 +_o suc 𝑧)))
12	11	breq2d 5178	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑦)) ↔ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧))))
13		suceq 6461	. . . . . . . . . 10 ⊢ (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
14	13	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑦 = suc 𝑧 → (𝐴 +_o suc 𝑦) = (𝐴 +_o suc suc 𝑧))
15	14	fveq2d 6924	. . . . . . . 8 ⊢ (𝑦 = suc 𝑧 → (𝐺‘(𝐴 +_o suc 𝑦)) = (𝐺‘(𝐴 +_o suc suc 𝑧)))
16	15	breq2d 5178	. . . . . . 7 ⊢ (𝑦 = suc 𝑧 → ((𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑦)) ↔ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc suc 𝑧))))
17		om2noseq.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ No )
18		om2noseq.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +_s 1_s )), 𝐶) ↾ ω))
19		om2noseq.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +_s 1_s )), 𝐶) “ ω))
20	17, 18, 19	om2noseqfo 28322	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:ω–onto→𝑍)
21		fof 6834	. . . . . . . . . . . . 13 ⊢ (𝐺:ω–onto→𝑍 → 𝐺:ω⟶𝑍)
22	20, 21	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ω⟶𝑍)
23	19, 17	noseqssno 28318	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ⊆ No )
24	22, 23	fssd 6764	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω⟶ No )
25	24	ffvelcdmda 7118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘𝐴) ∈ No )
26	25	addsridd 28016	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝐺‘𝐴) +_s 0_s ) = (𝐺‘𝐴))
27		0slt1s 27892	. . . . . . . . . 10 ⊢ 0_s <s 1_s
28		0sno 27889	. . . . . . . . . . . 12 ⊢ 0_s ∈ No
29	28	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → 0_s ∈ No )
30		1sno 27890	. . . . . . . . . . . 12 ⊢ 1_s ∈ No
31	30	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → 1_s ∈ No )
32	29, 31, 25	sltadd2d 28048	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ( 0_s <s 1_s ↔ ((𝐺‘𝐴) +_s 0_s ) <s ((𝐺‘𝐴) +_s 1_s )))
33	27, 32	mpbii 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝐺‘𝐴) +_s 0_s ) <s ((𝐺‘𝐴) +_s 1_s ))
34	26, 33	eqbrtrrd 5190	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘𝐴) <s ((𝐺‘𝐴) +_s 1_s ))
35		nnon 7909	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
36		oa1suc 8587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (𝐴 +_o 1_o) = suc 𝐴)
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (𝐴 +_o 1_o) = suc 𝐴)
38	37	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o 1_o)) = (𝐺‘suc 𝐴))
39	38	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘(𝐴 +_o 1_o)) = (𝐺‘suc 𝐴))
40	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → 𝐶 ∈ No )
41	18	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +_s 1_s )), 𝐶) ↾ ω))
42		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → 𝐴 ∈ ω)
43	40, 41, 42	om2noseqsuc 28321	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) +_s 1_s ))
44	39, 43	eqtrd 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘(𝐴 +_o 1_o)) = ((𝐺‘𝐴) +_s 1_s ))
45	34, 44	breqtrrd 5194	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o 1_o)))
46	25	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘𝐴) ∈ No )
47	24	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → 𝐺:ω⟶ No )
48		peano2 7929	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
49	48	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧))) → suc 𝑧 ∈ ω)
50		nnacl 8667	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ suc 𝑧 ∈ ω) → (𝐴 +_o suc 𝑧) ∈ ω)
51	42, 49, 50	syl2an 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐴 +_o suc 𝑧) ∈ ω)
52	47, 51	ffvelcdmd 7119	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘(𝐴 +_o suc 𝑧)) ∈ No )
53		peano2 7929	. . . . . . . . . . . . . 14 ⊢ (suc 𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
54	48, 53	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
55	54	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧))) → suc suc 𝑧 ∈ ω)
56		nnacl 8667	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ suc suc 𝑧 ∈ ω) → (𝐴 +_o suc suc 𝑧) ∈ ω)
57	42, 55, 56	syl2an 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐴 +_o suc suc 𝑧) ∈ ω)
58	47, 57	ffvelcdmd 7119	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘(𝐴 +_o suc suc 𝑧)) ∈ No )
59		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))
60	52	addsridd 28016	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → ((𝐺‘(𝐴 +_o suc 𝑧)) +_s 0_s ) = (𝐺‘(𝐴 +_o suc 𝑧)))
61	28	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → 0_s ∈ No )
62	30	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → 1_s ∈ No )
63	61, 62, 52	sltadd2d 28048	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → ( 0_s <s 1_s ↔ ((𝐺‘(𝐴 +_o suc 𝑧)) +_s 0_s ) <s ((𝐺‘(𝐴 +_o suc 𝑧)) +_s 1_s )))
64	27, 63	mpbii 233	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → ((𝐺‘(𝐴 +_o suc 𝑧)) +_s 0_s ) <s ((𝐺‘(𝐴 +_o suc 𝑧)) +_s 1_s ))
65	60, 64	eqbrtrrd 5190	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘(𝐴 +_o suc 𝑧)) <s ((𝐺‘(𝐴 +_o suc 𝑧)) +_s 1_s ))
66		nnasuc 8662	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ suc 𝑧 ∈ ω) → (𝐴 +_o suc suc 𝑧) = suc (𝐴 +_o suc 𝑧))
67	66	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ suc 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc suc 𝑧)) = (𝐺‘suc (𝐴 +_o suc 𝑧)))
68	42, 49, 67	syl2an 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘(𝐴 +_o suc suc 𝑧)) = (𝐺‘suc (𝐴 +_o suc 𝑧)))
69	17	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → 𝐶 ∈ No )
70	18	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +_s 1_s )), 𝐶) ↾ ω))
71	69, 70, 51	om2noseqsuc 28321	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘suc (𝐴 +_o suc 𝑧)) = ((𝐺‘(𝐴 +_o suc 𝑧)) +_s 1_s ))
72	68, 71	eqtrd 2780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘(𝐴 +_o suc suc 𝑧)) = ((𝐺‘(𝐴 +_o suc 𝑧)) +_s 1_s ))
73	65, 72	breqtrrd 5194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘(𝐴 +_o suc 𝑧)) <s (𝐺‘(𝐴 +_o suc suc 𝑧)))
74	46, 52, 58, 59, 73	slttrd 27822	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc suc 𝑧)))
75	74	expr 456	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ 𝑧 ∈ ω) → ((𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)) → (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc suc 𝑧))))
76	75	expcom 413	. . . . . . 7 ⊢ (𝑧 ∈ ω → ((𝜑 ∧ 𝐴 ∈ ω) → ((𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)) → (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc suc 𝑧)))))
77	8, 12, 16, 45, 76	finds2 7938	. . . . . 6 ⊢ (𝑦 ∈ ω → ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑦))))
78	77	impcom 407	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ 𝑦 ∈ ω) → (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑦)))
79		fveq2 6920	. . . . . 6 ⊢ ((𝐴 +_o suc 𝑦) = 𝐵 → (𝐺‘(𝐴 +_o suc 𝑦)) = (𝐺‘𝐵))
80	79	breq2d 5178	. . . . 5 ⊢ ((𝐴 +_o suc 𝑦) = 𝐵 → ((𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑦)) ↔ (𝐺‘𝐴) <s (𝐺‘𝐵)))
81	78, 80	syl5ibcom 245	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝐴 +_o suc 𝑦) = 𝐵 → (𝐺‘𝐴) <s (𝐺‘𝐵)))
82	81	rexlimdva 3161	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (∃𝑦 ∈ ω (𝐴 +_o suc 𝑦) = 𝐵 → (𝐺‘𝐴) <s (𝐺‘𝐵)))
83	82	adantrr 716	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (∃𝑦 ∈ ω (𝐴 +_o suc 𝑦) = 𝐵 → (𝐺‘𝐴) <s (𝐺‘𝐵)))
84	2, 83	sylbid 240	1 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) <s (𝐺‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 Vcvv 3488 ∅c0 4352 class class class wbr 5166 ↦ cmpt 5249 ↾ cres 5702 “ cima 5703 Oncon0 6395 suc csuc 6397 ⟶wf 6569 –onto→wfo 6571 ‘cfv 6573 (class class class)co 7448 ωcom 7903 reccrdg 8465 1_oc1o 8515 +_o coa 8519 No csur 27702 <s cslt 27703 0_s c0s 27885 1_s c1s 27886 +_s cadds 28010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-nadd 8722 df-no 27705 df-slt 27706 df-bday 27707 df-sle 27808 df-sslt 27844 df-scut 27846 df-0s 27887 df-1s 27888 df-made 27904 df-old 27905 df-left 27907 df-right 27908 df-norec2 28000 df-adds 28011

This theorem is referenced by: om2noseqlt2 28324 om2noseqf1o 28325

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