Theorem negsproplem2 28124

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 28118	. . . 4 ⊢ -us Fn No
2		fnfun 6623	. . . 4 ⊢ ( -us Fn No → Fun -us )
3	1, 2	ax-mp 5	. . 3 ⊢ Fun -us
4		fvex 6882	. . . 4 ⊢ ( R ‘𝐴) ∈ V
5	4	funimaex 6611	. . 3 ⊢ (Fun -us → ( -us “ ( R ‘𝐴)) ∈ V)
6	3, 5	mp1i 13	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) ∈ V)
7		fvex 6882	. . . 4 ⊢ ( L ‘𝐴) ∈ V
8	7	funimaex 6611	. . 3 ⊢ (Fun -us → ( -us “ ( L ‘𝐴)) ∈ V)
9	3, 8	mp1i 13	. 2 ⊢ (𝜑 → ( -us “ ( L ‘𝐴)) ∈ V)
10		rightssold 27967	. . . 4 ⊢ ( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
11		imass2 6093	. . . 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 484	. . . . . . 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 27936	. . . . . . . . 9 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
16	15	sseli 3934	. . . . . . . 8 ⊢ (𝑎 ∈ ( O ‘( bday ‘𝐴)) → 𝑎 ∈ No )
17	16	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → 𝑎 ∈ No )
18		0no 27904	. . . . . . . 8 ⊢ 0s ∈ No
19	18	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → 0s ∈ No )
20		bday0 27906	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
21	20	uneq2i 4120	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∪ ( bday ‘ 0s )) = (( bday ‘𝑎) ∪ ∅)
22		un0 4350	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∪ ∅) = ( bday ‘𝑎)
23	21, 22	eqtri 2787	. . . . . . . 8 ⊢ (( bday ‘𝑎) ∪ ( bday ‘ 0s )) = ( bday ‘𝑎)
24		oldbdayim 27984	. . . . . . . . . 10 ⊢ (𝑎 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑎) ∈ ( bday ‘𝐴))
25	24	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑎) ∈ ( bday ‘𝐴))
26		elun1 4136	. . . . . . . . 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 2868	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → (( bday ‘𝑎) ∪ ( bday ‘ 0s )) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
29	14, 17, 19, 28	negsproplem1 28123	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → (( -us ‘𝑎) ∈ No ∧ (𝑎 <s 0s → ( -us ‘ 0s ) <s ( -us ‘𝑎))))
30	29	simpld 498	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ( -us ‘𝑎) ∈ No )
31	30	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ ( O ‘( bday ‘𝐴))( -us ‘𝑎) ∈ No )
32	1	fndmi 6627	. . . . . 6 ⊢ dom -us = No
33	15, 32	sseqtrri 3987	. . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ⊆ dom -us
34		funimass4 6933	. . . . 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 702	. . . 4 ⊢ (( -us “ ( O ‘( bday ‘𝐴))) ⊆ No ↔ ∀𝑎 ∈ ( O ‘( bday ‘𝐴))( -us ‘𝑎) ∈ No )
36	31, 35	sylibr 236	. . 3 ⊢ (𝜑 → ( -us “ ( O ‘( bday ‘𝐴))) ⊆ No )
37	12, 36	sstrid 3949	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) ⊆ No )
38		leftssold 27966	. . . 4 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
39		imass2 6093	. . . 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 3949	. 2 ⊢ (𝜑 → ( -us “ ( L ‘𝐴)) ⊆ No )
42		rightssno 27969	. . . . . . 7 ⊢ ( R ‘𝐴) ⊆ No
43		fvelimab 6941	. . . . . . 7 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎))
44	1, 42, 43	mp2an 702	. . . . . 6 ⊢ (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎)
45		leftssno 27968	. . . . . . 7 ⊢ ( L ‘𝐴) ⊆ No
46		fvelimab 6941	. . . . . . 7 ⊢ (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏))
47	1, 45, 46	mp2an 702	. . . . . 6 ⊢ (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏)
48	44, 47	anbi12i 637	. . . . 5 ⊢ ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎 ∧ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏))
49		reeanv 3236	. . . . 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 280	. . . 4 ⊢ ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏))
51		lltr 27957	. . . . . . . . 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 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( L ‘𝐴))
54		simprl 780	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( R ‘𝐴))
55	52, 53, 54	sltssepcd 27867	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 <s 𝑥𝑅)
56	13	adantr 484	. . . . . . . . 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 3934	. . . . . . . . . 10 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ No )
58	57	ad2antll 739	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ No )
59	42	sseli 3934	. . . . . . . . . . 11 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 ∈ No )
60	59	adantr 484	. . . . . . . . . 10 ⊢ ((𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴)) → 𝑥𝑅 ∈ No )
61	60	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ No )
62	38	sseli 3934	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)))
63	62	ad2antll 739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)))
64		oldbdayim 27984	. . . . . . . . . . . 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 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)))
68	67	adantrr 727	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)))
69		oldbdayim 27984	. . . . . . . . . . . 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		bdayon 27847	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑥𝐿) ∈ On
72		bdayon 27847	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑥𝑅) ∈ On
73		bdayon 27847	. . . . . . . . . . . 12 ⊢ ( bday ‘𝐴) ∈ On
74		onunel 6455	. . . . . . . . . . . 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 1484	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑥𝐿) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑥𝑅) ∈ ( bday ‘𝐴)))
76	65, 70, 75	sylanbrc 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴))
77		elun1 4136	. . . . . . . . . 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 28123	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( -us ‘𝑥𝐿) ∈ No ∧ (𝑥𝐿 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿))))
80	79	simprd 499	. . . . . . 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 5107	. . . . . 6 ⊢ ((( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → (( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿) ↔ 𝑎 <s 𝑏))
83	81, 82	syl5ibcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ((( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
84	83	rexlimdvva 3221	. . . 4 ⊢ (𝜑 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
85	50, 84	biimtrid 244	. . 3 ⊢ (𝜑 → ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏))
86	85	3impib 1130	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏)
87	6, 9, 37, 41, 86	sltsd 27863	1 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088  Vcvv 3456   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287   class class class wbr 5102  dom cdm 5649   “ cima 5652  Oncon0 6348  Fun wfun 6517   Fn wfn 6518  ‘cfv 6523   No csur 27706   <s clts 27707   bday cbday 27708   <<s cslts 27852   0_s c0s 27900   O cold 27918   L cleft 27920   R cright 27921   -u_s cnegs 28114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-1o 8439  df-2o 8440  df-no 27709  df-lts 27710  df-bday 27711  df-slts 27853  df-cuts 27855  df-0s 27902  df-made 27922  df-old 27923  df-left 27925  df-right 27926  df-norec 28033  df-negs 28116

This theorem is referenced by:  negsproplem3  28125