Theorem negsproplem2 27742

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 27737	. . . 4 ⊢ -us Fn No
2		fnfun 6648	. . . 4 ⊢ ( -us Fn No → Fun -us )
3	1, 2	ax-mp 5	. . 3 ⊢ Fun -us
4		fvex 6903	. . . 4 ⊢ ( R ‘𝐴) ∈ V
5	4	funimaex 6635	. . 3 ⊢ (Fun -us → ( -us “ ( R ‘𝐴)) ∈ V)
6	3, 5	mp1i 13	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) ∈ V)
7		fvex 6903	. . . 4 ⊢ ( L ‘𝐴) ∈ V
8	7	funimaex 6635	. . 3 ⊢ (Fun -us → ( -us “ ( L ‘𝐴)) ∈ V)
9	3, 8	mp1i 13	. 2 ⊢ (𝜑 → ( -us “ ( L ‘𝐴)) ∈ V)
10		rightssold 27611	. . . 4 ⊢ ( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
11		imass2 6100	. . . 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 479	. . . . . . 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 27593	. . . . . . . . 9 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
16	15	sseli 3977	. . . . . . . 8 ⊢ (𝑎 ∈ ( O ‘( bday ‘𝐴)) → 𝑎 ∈ No )
17	16	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → 𝑎 ∈ No )
18		0sno 27564	. . . . . . . 8 ⊢ 0s ∈ No
19	18	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → 0s ∈ No )
20		bday0s 27566	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
21	20	uneq2i 4159	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∪ ( bday ‘ 0s )) = (( bday ‘𝑎) ∪ ∅)
22		un0 4389	. . . . . . . . 9 ⊢ (( bday ‘𝑎) ∪ ∅) = ( bday ‘𝑎)
23	21, 22	eqtri 2758	. . . . . . . 8 ⊢ (( bday ‘𝑎) ∪ ( bday ‘ 0s )) = ( bday ‘𝑎)
24		oldbdayim 27620	. . . . . . . . . 10 ⊢ (𝑎 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑎) ∈ ( bday ‘𝐴))
25	24	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ( bday ‘𝑎) ∈ ( bday ‘𝐴))
26		elun1 4175	. . . . . . . . 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 2835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → (( bday ‘𝑎) ∪ ( bday ‘ 0s )) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
29	14, 17, 19, 28	negsproplem1 27741	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → (( -us ‘𝑎) ∈ No ∧ (𝑎 <s 0s → ( -us ‘ 0s ) <s ( -us ‘𝑎))))
30	29	simpld 493	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ( O ‘( bday ‘𝐴))) → ( -us ‘𝑎) ∈ No )
31	30	ralrimiva 3144	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ ( O ‘( bday ‘𝐴))( -us ‘𝑎) ∈ No )
32	1	fndmi 6652	. . . . . 6 ⊢ dom -us = No
33	15, 32	sseqtrri 4018	. . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ⊆ dom -us
34		funimass4 6955	. . . . 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 688	. . . 4 ⊢ (( -us “ ( O ‘( bday ‘𝐴))) ⊆ No ↔ ∀𝑎 ∈ ( O ‘( bday ‘𝐴))( -us ‘𝑎) ∈ No )
36	31, 35	sylibr 233	. . 3 ⊢ (𝜑 → ( -us “ ( O ‘( bday ‘𝐴))) ⊆ No )
37	12, 36	sstrid 3992	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) ⊆ No )
38		leftssold 27610	. . . 4 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
39		imass2 6100	. . . 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 3992	. 2 ⊢ (𝜑 → ( -us “ ( L ‘𝐴)) ⊆ No )
42		rightssno 27613	. . . . . . 7 ⊢ ( R ‘𝐴) ⊆ No
43		fvelimab 6963	. . . . . . 7 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎))
44	1, 42, 43	mp2an 688	. . . . . 6 ⊢ (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎)
45		leftssno 27612	. . . . . . 7 ⊢ ( L ‘𝐴) ⊆ No
46		fvelimab 6963	. . . . . . 7 ⊢ (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏))
47	1, 45, 46	mp2an 688	. . . . . 6 ⊢ (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏)
48	44, 47	anbi12i 625	. . . . 5 ⊢ ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)( -us ‘𝑥𝑅) = 𝑎 ∧ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us ‘𝑥𝐿) = 𝑏))
49		reeanv 3224	. . . . 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 277	. . . 4 ⊢ ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏))
51		lltropt 27604	. . . . . . . . 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 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( L ‘𝐴))
54		simprl 767	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( R ‘𝐴))
55	52, 53, 54	ssltsepcd 27532	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 <s 𝑥𝑅)
56	13	adantr 479	. . . . . . . . 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 3977	. . . . . . . . . 10 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ No )
58	57	ad2antll 725	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ No )
59	42	sseli 3977	. . . . . . . . . . 11 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 ∈ No )
60	59	adantr 479	. . . . . . . . . 10 ⊢ ((𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴)) → 𝑥𝑅 ∈ No )
61	60	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ No )
62	38	sseli 3977	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)))
63	62	ad2antll 725	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)))
64		oldbdayim 27620	. . . . . . . . . . . 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 3981	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)))
68	67	adantrr 713	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)))
69		oldbdayim 27620	. . . . . . . . . . . 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 27514	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑥𝐿) ∈ On
72		bdayelon 27514	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑥𝑅) ∈ On
73		bdayelon 27514	. . . . . . . . . . . 12 ⊢ ( bday ‘𝐴) ∈ On
74		onunel 6468	. . . . . . . . . . . 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 1459	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑥𝐿) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑥𝑅) ∈ ( bday ‘𝐴)))
76	65, 70, 75	sylanbrc 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( bday ‘𝑥𝐿) ∪ ( bday ‘𝑥𝑅)) ∈ ( bday ‘𝐴))
77		elun1 4175	. . . . . . . . . 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 27741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( -us ‘𝑥𝐿) ∈ No ∧ (𝑥𝐿 <s 𝑥𝑅 → ( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿))))
80	79	simprd 494	. . . . . . 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 5152	. . . . . 6 ⊢ ((( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → (( -us ‘𝑥𝑅) <s ( -us ‘𝑥𝐿) ↔ 𝑎 <s 𝑏))
83	81, 82	syl5ibcom 244	. . . . 5 ⊢ ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ((( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
84	83	rexlimdvva 3209	. . . 4 ⊢ (𝜑 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us ‘𝑥𝑅) = 𝑎 ∧ ( -us ‘𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
85	50, 84	biimtrid 241	. . 3 ⊢ (𝜑 → ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏))
86	85	3impib 1114	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏)
87	6, 9, 37, 41, 86	ssltd 27529	1 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068  Vcvv 3472   ∪ cun 3945   ⊆ wss 3947  ∅c0 4321   class class class wbr 5147  dom cdm 5675   “ cima 5678  Oncon0 6363  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542   No csur 27379   <s cslt 27380   bday cbday 27381   <<s csslt 27518   0_s c0s 27560   O cold 27575   L cleft 27577   R cright 27578   -u_s cnegs 27733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-1o 8468  df-2o 8469  df-no 27382  df-slt 27383  df-bday 27384  df-sslt 27519  df-scut 27521  df-0s 27562  df-made 27579  df-old 27580  df-left 27582  df-right 27583  df-norec 27660  df-negs 27735

This theorem is referenced by:  negsproplem3  27743