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Theorem om2noseqlt 28260

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 8660	. . 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 6431	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
4		df-1o 8489	. . . . . . . . . . 11 ⊢ 1_o = suc ∅
5	3, 4	eqtr4di 2787	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → suc 𝑦 = 1_o)
6	5	oveq2d 7430	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝐴 +_o suc 𝑦) = (𝐴 +_o 1_o))
7	6	fveq2d 6891	. . . . . . . 8 ⊢ (𝑦 = ∅ → (𝐺‘(𝐴 +_o suc 𝑦)) = (𝐺‘(𝐴 +_o 1_o)))
8	7	breq2d 5137	. . . . . . 7 ⊢ (𝑦 = ∅ → ((𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑦)) ↔ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o 1_o))))
9		suceq 6431	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
10	9	oveq2d 7430	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝐴 +_o suc 𝑦) = (𝐴 +_o suc 𝑧))
11	10	fveq2d 6891	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝐺‘(𝐴 +_o suc 𝑦)) = (𝐺‘(𝐴 +_o suc 𝑧)))
12	11	breq2d 5137	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑦)) ↔ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧))))
13		suceq 6431	. . . . . . . . . 10 ⊢ (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
14	13	oveq2d 7430	. . . . . . . . 9 ⊢ (𝑦 = suc 𝑧 → (𝐴 +_o suc 𝑦) = (𝐴 +_o suc suc 𝑧))
15	14	fveq2d 6891	. . . . . . . 8 ⊢ (𝑦 = suc 𝑧 → (𝐺‘(𝐴 +_o suc 𝑦)) = (𝐺‘(𝐴 +_o suc suc 𝑧)))
16	15	breq2d 5137	. . . . . . 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 28259	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:ω–onto→𝑍)
21		fof 6801	. . . . . . . . . . . . 13 ⊢ (𝐺:ω–onto→𝑍 → 𝐺:ω⟶𝑍)
22	20, 21	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ω⟶𝑍)
23	19, 17	noseqssno 28255	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ⊆ No )
24	22, 23	fssd 6734	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω⟶ No )
25	24	ffvelcdmda 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘𝐴) ∈ No )
26	25	addsridd 27953	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝐺‘𝐴) +_s 0_s ) = (𝐺‘𝐴))
27		0slt1s 27829	. . . . . . . . . 10 ⊢ 0_s <s 1_s
28		0sno 27826	. . . . . . . . . . . 12 ⊢ 0_s ∈ No
29	28	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → 0_s ∈ No )
30		1sno 27827	. . . . . . . . . . . 12 ⊢ 1_s ∈ No
31	30	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → 1_s ∈ No )
32	29, 31, 25	sltadd2d 27985	. . . . . . . . . 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 5149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘𝐴) <s ((𝐺‘𝐴) +_s 1_s ))
35		nnon 7876	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
36		oa1suc 8552	. . . . . . . . . . . 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 6891	. . . . . . . . . 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 28258	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) +_s 1_s ))
44	39, 43	eqtrd 2769	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘(𝐴 +_o 1_o)) = ((𝐺‘𝐴) +_s 1_s ))
45	34, 44	breqtrrd 5153	. . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → 𝐺:ω⟶ No )
48		peano2 7895	. . . . . . . . . . . . 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 8632	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ suc 𝑧 ∈ ω) → (𝐴 +_o suc 𝑧) ∈ ω)
51	42, 49, 50	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐴 +_o suc 𝑧) ∈ ω)
52	47, 51	ffvelcdmd 7086	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐺‘(𝐴 +_o suc 𝑧)) ∈ No )
53		peano2 7895	. . . . . . . . . . . . . 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 8632	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ suc suc 𝑧 ∈ ω) → (𝐴 +_o suc suc 𝑧) ∈ ω)
57	42, 55, 56	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → (𝐴 +_o suc suc 𝑧) ∈ ω)
58	47, 57	ffvelcdmd 7086	. . . . . . . . . 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 27953	. . . . . . . . . . . 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 27985	. . . . . . . . . . . . 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 5149	. . . . . . . . . . 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 8627	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ suc 𝑧 ∈ ω) → (𝐴 +_o suc suc 𝑧) = suc (𝐴 +_o suc 𝑧))
67	66	fveq2d 6891	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ suc 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc suc 𝑧)) = (𝐺‘suc (𝐴 +_o suc 𝑧)))
68	42, 49, 67	syl2an 596	. . . . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ω) ∧ (𝑧 ∈ ω ∧ (𝐺‘𝐴) <s (𝐺‘(𝐴 +_o suc 𝑧)))) → 𝐶 ∈ No )
70	18	ad2antrr 726	. . . . . . . . . . . . 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 28258	. . . . . . . . . . . 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 2769	. . . . . . . . . . 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 5153	. . . . . . . . . 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 27759	. . . . . . . . 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 7903	. . . . . 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 6887	. . . . . 6 ⊢ ((𝐴 +_o suc 𝑦) = 𝐵 → (𝐺‘(𝐴 +_o suc 𝑦)) = (𝐺‘𝐵))
80	79	breq2d 5137	. . . . 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 3142	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (∃𝑦 ∈ ω (𝐴 +_o suc 𝑦) = 𝐵 → (𝐺‘𝐴) <s (𝐺‘𝐵)))
83	82	adantrr 717	. 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 1539 ∈ wcel 2107 ∃wrex 3059 Vcvv 3464 ∅c0 4315 class class class wbr 5125 ↦ cmpt 5207 ↾ cres 5669 “ cima 5670 Oncon0 6365 suc csuc 6367 ⟶wf 6538 –onto→wfo 6540 ‘cfv 6542 (class class class)co 7414 ωcom 7870 reccrdg 8432 1_oc1o 8482 +_o coa 8486 No csur 27639 <s cslt 27640 0_s c0s 27822 1_s c1s 27823 +_s cadds 27947

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5261 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-pss 3953 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-tp 4613 df-op 4615 df-ot 4617 df-uni 4890 df-int 4929 df-iun 4975 df-br 5126 df-opab 5188 df-mpt 5208 df-tr 5242 df-id 5560 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6303 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7871 df-1st 7997 df-2nd 7998 df-frecs 8289 df-wrecs 8320 df-recs 8394 df-rdg 8433 df-1o 8489 df-2o 8490 df-oadd 8493 df-nadd 8687 df-no 27642 df-slt 27643 df-bday 27644 df-sle 27745 df-sslt 27781 df-scut 27783 df-0s 27824 df-1s 27825 df-made 27841 df-old 27842 df-left 27844 df-right 27845 df-norec2 27937 df-adds 27948

This theorem is referenced by: om2noseqlt2 28261 om2noseqf1o 28262

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