Theorem negsproplem2 28062

Description: Lemma for surreal negation. Show that the cut that defines negation is legitimate. (Contributed by Scott Fenton, 2-Feb-2025.)

Hypotheses

Ref	Expression
negsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
negsproplem2.1	⊢ (𝜑 → 𝐴 ∈ No )

Assertion

Ref	Expression
negsproplem2	⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem negsproplem2

Dummy variables 𝑎 𝑏 𝑥_𝐿 𝑥_𝑅 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		negsfn 28056	. . . 4 ⊢ -us Fn No
2		fnfun 6667	. . . 4 ⊢ ( -us Fn No → Fun -us )
3	1, 2	ax-mp 5	. . 3 ⊢ Fun -us
4		fvex 6918	. . . 4 ⊢ ( R ‘𝐴) ∈ V
5	4	funimaex 6654	. . 3 ⊢ (Fun -us → ( -us “ ( R ‘𝐴)) ∈ V)
6	3, 5	mp1i 13	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) ∈ V)
7		fvex 6918	. . . 4 ⊢ ( L ‘𝐴) ∈ V
8	7	funimaex 6654	. . 3 ⊢ (Fun -us → ( -us “ ( L ‘𝐴)) ∈ V)
9	3, 8	mp1i 13	. 2 ⊢ (𝜑 → ( -us “ ( L ‘𝐴)) ∈ V)
10		rightssold 27919	. . . 4 ⊢ ( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
11		imass2 6119	. . . 4 ⊢ (( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴)) → ( -us “ ( R ‘𝐴)) ⊆ ( -us “ ( O ‘( bday ‘𝐴))))
12	10, 11	ax-mp 5	. . 3 ⊢ ( -us “ ( R ‘𝐴)) ⊆ ( -us “ ( O ‘( bday ‘𝐴)))
13		negsproplem.1	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
14	13	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
15		oldssno 27901	. . . . . . . . 9 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
16	15	sseli 3978	. . . . . . . 8 ⊢ (𝑎 ∈ ( O ‘( bday ‘𝐴)) → 𝑎 ∈ No )
17	16	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → 𝑎 ∈ No )
18		0sno 27872	. . . . . . . 8 ⊢ 0s ∈ No
19	18	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → 0s ∈ No )
20		bday0s 27874	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
21	20	uneq2i 4164	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∪ ( bday ‘ 0s )) = (( bday ‘𝑎) ∪ ∅)
22		un0 4393	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∪ ∅) = ( bday ‘𝑎)
23	21, 22	eqtri 2764	. . . . . . . 8 ⊢ (( bday ‘𝑎) ∪ ( bday ‘ 0s )) = ( bday ‘𝑎)
24		oldbdayim 27928	. . . . . . . . . 10 ⊢ (𝑎 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑎) ∈ ( bday ‘𝐴))
25	24	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑎) ∈ ( bday ‘𝐴))
26		elun1 4181	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∈ ( bday ‘𝐴) → ( bday ‘𝑎) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑎) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
28	23, 27	eqeltrid 2844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → (( bday ‘𝑎) ∪ ( bday ‘ 0s )) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
29	14, 17, 19, 28	negsproplem1 28061	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → (( -us ‘𝑎) ∈ No ∧ (𝑎 <s 0s → ( -us ‘ 0s ) <s ( -us ‘𝑎))))
30	29	simpld 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ( -us ‘𝑎) ∈ No )
31	30	ralrimiva 3145	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ ( O ‘( bday ‘𝐴))( -us ‘𝑎) ∈ No )
32	1	fndmi 6671	. . . . . 6 ⊢ dom -us = No
33	15, 32	sseqtrri 4032	. . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ⊆ dom -us
34		funimass4 6972	. . . . 5 ⊢ ((Fun -us ∧ ( O ‘( bday ‘𝐴)) ⊆ dom -us ) → (( -us “ ( O ‘( bday ‘𝐴))) ⊆ No ↔ ∀𝑎 ∈ ( O ‘( bday ‘𝐴))( -us ‘𝑎) ∈ No ))
35	3, 33, 34	mp2an 692	. . . 4 ⊢ (( -us “ ( O ‘( bday ‘𝐴))) ⊆ No ↔ ∀𝑎 ∈ ( O ‘( bday ‘𝐴))( -us ‘𝑎) ∈ No )
36	31, 35	sylibr 234	. . 3 ⊢ (𝜑 → ( -us “ ( O ‘( bday ‘𝐴))) ⊆ No )
37	12, 36	sstrid 3994	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) ⊆ No )
38		leftssold 27918	. . . 4 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
39		imass2 6119	. . . 4 ⊢ (( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴)) → ( -us “ ( L ‘𝐴)) ⊆ ( -us “ ( O ‘( bday ‘𝐴))))
40	38, 39	ax-mp 5	. . 3 ⊢ ( -us “ ( L ‘𝐴)) ⊆ ( -us “ ( O ‘( bday ‘𝐴)))
41	40, 36	sstrid 3994	. 2 ⊢ (𝜑 → ( -us “ ( L ‘𝐴)) ⊆ No )
42		rightssno 27921	. . . . . . 7 ⊢ ( R ‘𝐴) ⊆ No
43		fvelimab 6980	. . . . . . 7 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎))
44	1, 42, 43	mp2an 692	. . . . . 6 ⊢ (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎)
45		leftssno 27920	. . . . . . 7 ⊢ ( L ‘𝐴) ⊆ No
46		fvelimab 6980	. . . . . . 7 ⊢ (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏))
47	1, 45, 46	mp2an 692	. . . . . 6 ⊢ (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏)
48	44, 47	anbi12i 628	. . . . 5 ⊢ ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎 ∧ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏))
49		reeanv 3228	. . . . 5 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎 ∧ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏))
50	48, 49	bitr4i 278	. . . 4 ⊢ ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏))
51		lltropt 27912	. . . . . . . . 9 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
52	51	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( L ‘𝐴) <<s ( R ‘𝐴))
53		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( L ‘𝐴))
54		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( R ‘𝐴))
55	52, 53, 54	ssltsepcd 27840	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 <s 𝑥𝑅)
56	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
57	45	sseli 3978	. . . . . . . . . 10 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ No )
58	57	ad2antll 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ No )
59	42	sseli 3978	. . . . . . . . . . 11 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 ∈ No )
60	59	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴)) → 𝑥𝑅 ∈ No )
61	60	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ No )
62	38	sseli 3978	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)))
63	62	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)))
64		oldbdayim 27928	. . . . . . . . . . . 12 ⊢ (𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑥𝐿) ∈ ( bday ‘𝐴))
65	63, 64	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( bday ‘𝑥𝐿) ∈ ( bday ‘𝐴))
66	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴)))
67	66	sselda 3982	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)))
68	67	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)))
69		oldbdayim 27928	. . . . . . . . . . . 12 ⊢ (𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑥𝑅) ∈ ( bday ‘𝐴))
70	68, 69	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( bday ‘𝑥𝑅) ∈ ( bday ‘𝐴))
71		bdayelon 27822	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑥𝐿) ∈ On
72		bdayelon 27822	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑥𝑅) ∈ On
73		bdayelon 27822	. . . . . . . . . . . 12 ⊢ ( bday ‘𝐴) ∈ On
74		onunel 6488	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑥𝐿) ∈ On ∧ ( bday ‘𝑥𝑅) ∈ On ∧ ( bday ‘𝐴) ∈ On) → ((( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑥𝐿) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑥𝑅) ∈ ( bday ‘𝐴))))
75	71, 72, 73, 74	mp3an 1462	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑥𝐿) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑥𝑅) ∈ ( bday ‘𝐴)))
76	65, 70, 75	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴))
77		elun1 4181	. . . . . . . . . 10 ⊢ ((( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴) → (( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
78	76, 77	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
79	56, 58, 61, 78	negsproplem1 28061	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( -us ‘𝑥𝐿) ∈ No ∧ (𝑥𝐿 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿))))
80	79	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (𝑥𝐿 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿)))
81	55, 80	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿))
82		breq12 5147	. . . . . 6 ⊢ ((( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → (( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿) ↔ 𝑎 <s 𝑏))
83	81, 82	syl5ibcom 245	. . . . 5 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ((( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
84	83	rexlimdvva 3212	. . . 4 ⊢ (𝜑 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
85	50, 84	biimtrid 242	. . 3 ⊢ (𝜑 → ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏))
86	85	3impib 1116	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏)
87	6, 9, 37, 41, 86	ssltd 27837	1 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∪ cun 3948   ⊆ wss 3950  ∅c0 4332   class class class wbr 5142  dom cdm 5684   “ cima 5687  Oncon0 6383  Fun wfun 6554   Fn wfn 6555  ‘cfv 6560   No csur 27685   <s cslt 27686   bday cbday 27687   <<s csslt 27826   0_s c0s 27868   O cold 27883   L cleft 27885   R cright 27886   -u_s cnegs 28052

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 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689  df-bday 27690  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-negs 28054

This theorem is referenced by:  negsproplem3  28063