Theorem negsproplem2 28290

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 28284	. . . 4 ⊢ -us Fn No
2		fnfun 6636	. . . 4 ⊢ ( -us Fn No → Fun -us )
3	1, 2	ax-mp 5	. . 3 ⊢ Fun -us
4		fvex 6895	. . . 4 ⊢ ( R ‘𝐴) ∈ V
5	4	funimaex 6624	. . 3 ⊢ (Fun -us → ( -us “ ( R ‘𝐴)) ∈ V)
6	3, 5	mp1i 14	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) ∈ V)
7		fvex 6895	. . . 4 ⊢ ( L ‘𝐴) ∈ V
8	7	funimaex 6624	. . 3 ⊢ (Fun -us → ( -us “ ( L ‘𝐴)) ∈ V)
9	3, 8	mp1i 14	. 2 ⊢ (𝜑 → ( -us “ ( L ‘𝐴)) ∈ V)
10		rightssold 28133	. . . 4 ⊢ ( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
11		imass2 6102	. . . 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 486	. . . . . . 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 28102	. . . . . . . . 9 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
16	15	sseli 3930	. . . . . . . 8 ⊢ (𝑎 ∈ ( O ‘( bday ‘𝐴)) → 𝑎 ∈ No )
17	16	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → 𝑎 ∈ No )
18		0no 28070	. . . . . . . 8 ⊢ 0s ∈ No
19	18	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → 0s ∈ No )
20		bday0 28072	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
21	20	uneq2i 4115	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∪ ( bday ‘ 0s )) = (( bday ‘𝑎) ∪ ∅)
22		un0 4347	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∪ ∅) = ( bday ‘𝑎)
23	21, 22	eqtri 2785	. . . . . . . 8 ⊢ (( bday ‘𝑎) ∪ ( bday ‘ 0s )) = ( bday ‘𝑎)
24		oldbdayim 28150	. . . . . . . . . 10 ⊢ (𝑎 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑎) ∈ ( bday ‘𝐴))
25	24	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑎) ∈ ( bday ‘𝐴))
26		elun1 4131	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∈ ( bday ‘𝐴) → ( bday ‘𝑎) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
27	25, 26	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑎) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
28	23, 27	eqeltrid 2866	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → (( bday ‘𝑎) ∪ ( bday ‘ 0s )) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
29	14, 17, 19, 28	negsproplem1 28289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → (( -us ‘𝑎) ∈ No ∧ (𝑎 <s 0s → ( -us ‘ 0s ) <s ( -us ‘𝑎))))
30	29	simpld 500	. . . . 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 6640	. . . . . 6 ⊢ dom -us = No
33	15, 32	sseqtrri 3983	. . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ⊆ dom -us
34		funimass4 6946	. . . . 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 705	. . . 4 ⊢ (( -us “ ( O ‘( bday ‘𝐴))) ⊆ No ↔ ∀𝑎 ∈ ( O ‘( bday ‘𝐴))( -us ‘𝑎) ∈ No )
36	31, 35	sylibr 237	. . 3 ⊢ (𝜑 → ( -us “ ( O ‘( bday ‘𝐴))) ⊆ No )
37	12, 36	sstrid 3945	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) ⊆ No )
38		leftssold 28132	. . . 4 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
39		imass2 6102	. . . 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 3945	. 2 ⊢ (𝜑 → ( -us “ ( L ‘𝐴)) ⊆ No )
42		rightssno 28135	. . . . . . 7 ⊢ ( R ‘𝐴) ⊆ No
43		fvelimab 6954	. . . . . . 7 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎))
44	1, 42, 43	mp2an 705	. . . . . 6 ⊢ (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎)
45		leftssno 28134	. . . . . . 7 ⊢ ( L ‘𝐴) ⊆ No
46		fvelimab 6954	. . . . . . 7 ⊢ (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏))
47	1, 45, 46	mp2an 705	. . . . . 6 ⊢ (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏)
48	44, 47	anbi12i 640	. . . . 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 281	. . . 4 ⊢ ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏))
51		lltr 28123	. . . . . . . . 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 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( L ‘𝐴))
54		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( R ‘𝐴))
55	52, 53, 54	sltssepcd 28033	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 <s 𝑥𝑅)
56	13	adantr 486	. . . . . . . . 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 3930	. . . . . . . . . 10 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ No )
58	57	ad2antll 742	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ No )
59	42	sseli 3930	. . . . . . . . . . 11 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 ∈ No )
60	59	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴)) → 𝑥𝑅 ∈ No )
61	60	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ No )
62	38	sseli 3930	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)))
63	62	ad2antll 742	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)))
64		oldbdayim 28150	. . . . . . . . . . . 12 ⊢ (𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑥𝐿) ∈ ( bday ‘𝐴))
65	63, 64	syl 18	. . . . . . . . . . 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 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)))
68	67	adantrr 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)))
69		oldbdayim 28150	. . . . . . . . . . . 12 ⊢ (𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑥𝑅) ∈ ( bday ‘𝐴))
70	68, 69	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( bday ‘𝑥𝑅) ∈ ( bday ‘𝐴))
71		bdayon 28013	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑥𝐿) ∈ On
72		bdayon 28013	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑥𝑅) ∈ On
73		bdayon 28013	. . . . . . . . . . . 12 ⊢ ( bday ‘𝐴) ∈ On
74		onunel 6469	. . . . . . . . . . . 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 1490	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑥𝐿) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑥𝑅) ∈ ( bday ‘𝐴)))
76	65, 70, 75	sylanbrc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴))
77		elun1 4131	. . . . . . . . . 10 ⊢ ((( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴) → (( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
78	76, 77	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
79	56, 58, 61, 78	negsproplem1 28289	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( -us ‘𝑥𝐿) ∈ No ∧ (𝑥𝐿 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿))))
80	79	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (𝑥𝐿 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿)))
81	55, 80	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿))
82		breq12 5112	. . . . . 6 ⊢ ((( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → (( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿) ↔ 𝑎 <s 𝑏))
83	81, 82	syl5ibcom 248	. . . . 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 245	. . 3 ⊢ (𝜑 → ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏))
86	85	3impib 1134	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏)
87	6, 9, 37, 41, 86	sltsd 28029	1 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  ∅c0 4282   class class class wbr 5107  dom cdm 5659   “ cima 5662  Oncon0 6361  Fun wfun 6531   Fn wfn 6532  ‘cfv 6537   No csur 27872   <s clts 27873   bday cbday 27874   <<s cslts 28018   0_s c0s 28066   O cold 28084   L cleft 28086   R cright 28087   -u_s cnegs 28280

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8458  df-2o 8459  df-no 27875  df-lts 27876  df-bday 27877  df-slts 28019  df-cuts 28021  df-0s 28068  df-made 28088  df-old 28089  df-left 28091  df-right 28092  df-norec 28199  df-negs 28282

This theorem is used by:  negsproplem3  28291