Theorem negsid 28433

Description: Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.)

Assertion

Ref	Expression
negsid	⊢ (𝐴 ∈ No → (𝐴 +s ( -s‘𝐴)) = 0s )

Proof of Theorem negsid

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑝 𝑞 𝑥 𝑥_𝐿 𝑥_𝑅 𝑥_𝑂 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → 𝑥 = 𝑥𝑂)
2		fveq2 6886	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → ( -s‘𝑥) = ( -s‘𝑥𝑂))
3	1, 2	oveq12d 7439	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s ( -s‘𝑥)) = (𝑥𝑂 +s ( -s‘𝑥𝑂)))
4	3	eqeq1d 2762	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s ( -s‘𝑥)) = 0s ↔ (𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ))
5		id 23	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
6		fveq2 6886	. . . 4 ⊢ (𝑥 = 𝐴 → ( -s‘𝑥) = ( -s‘𝐴))
7	5, 6	oveq12d 7439	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -s‘𝑥)) = (𝐴 +s ( -s‘𝐴)))
8	7	eqeq1d 2762	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s ( -s‘𝑥)) = 0s ↔ (𝐴 +s ( -s‘𝐴)) = 0s ))
9		lltr 28254	. . . . . 6 ⊢ (L‘𝑥) <<s (R‘𝑥)
10	9	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (L‘𝑥) <<s (R‘𝑥))
11		negcut2 28432	. . . . . 6 ⊢ (𝑥 ∈ No → ( -s “ (R‘𝑥)) <<s ( -s “ (L‘𝑥)))
12	11	adantr 486	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ( -s “ (R‘𝑥)) <<s ( -s “ (L‘𝑥)))
13		lrcut 28296	. . . . . . 7 ⊢ (𝑥 ∈ No → ((L‘𝑥) |s (R‘𝑥)) = 𝑥)
14	13	adantr 486	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ((L‘𝑥) |s (R‘𝑥)) = 𝑥)
15	14	eqcomd 2766	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → 𝑥 = ((L‘𝑥) |s (R‘𝑥)))
16		negsval 28417	. . . . . 6 ⊢ (𝑥 ∈ No → ( -s‘𝑥) = (( -s “ (R‘𝑥)) |s ( -s “ (L‘𝑥))))
17	16	adantr 486	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ( -s‘𝑥) = (( -s “ (R‘𝑥)) |s ( -s “ (L‘𝑥))))
18	10, 12, 15, 17	addsunif 28394	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (𝑥 +s ( -s‘𝑥)) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑝 ∈ ( -s “ (R‘𝑥))𝑏 = (𝑥 +s 𝑝)}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑞 ∈ ( -s “ (L‘𝑥))𝑑 = (𝑥 +s 𝑞)})))
19		negsfn 28415	. . . . . . . . 9 ⊢ -s Fn No
20		rightssno 28266	. . . . . . . . 9 ⊢ (R‘𝑥) ⊆ No
21		oveq2 7429	. . . . . . . . . . 11 ⊢ (𝑝 = ( -s‘𝑥𝑅) → (𝑥 +s 𝑝) = (𝑥 +s ( -s‘𝑥𝑅)))
22	21	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑝 = ( -s‘𝑥𝑅) → (𝑏 = (𝑥 +s 𝑝) ↔ 𝑏 = (𝑥 +s ( -s‘𝑥𝑅))))
23	22	rexima 7245	. . . . . . . . 9 ⊢ (( -s Fn No ∧ (R‘𝑥) ⊆ No) → (∃𝑝 ∈ ( -s “ (R‘𝑥))𝑏 = (𝑥 +s 𝑝) ↔ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))))
24	19, 20, 23	mp2an 705	. . . . . . . 8 ⊢ (∃𝑝 ∈ ( -s “ (R‘𝑥))𝑏 = (𝑥 +s 𝑝) ↔ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅)))
25	24	abbii 2827	. . . . . . 7 ⊢ {𝑏 ∣ ∃𝑝 ∈ ( -s “ (R‘𝑥))𝑏 = (𝑥 +s 𝑝)} = {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}
26	25	uneq2i 4164	. . . . . 6 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑝 ∈ ( -s “ (R‘𝑥))𝑏 = (𝑥 +s 𝑝)}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))})
27		leftssno 28265	. . . . . . . . 9 ⊢ (L‘𝑥) ⊆ No
28		oveq2 7429	. . . . . . . . . . 11 ⊢ (𝑞 = ( -s‘𝑥𝐿) → (𝑥 +s 𝑞) = (𝑥 +s ( -s‘𝑥𝐿)))
29	28	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑞 = ( -s‘𝑥𝐿) → (𝑑 = (𝑥 +s 𝑞) ↔ 𝑑 = (𝑥 +s ( -s‘𝑥𝐿))))
30	29	rexima 7245	. . . . . . . . 9 ⊢ (( -s Fn No ∧ (L‘𝑥) ⊆ No) → (∃𝑞 ∈ ( -s “ (L‘𝑥))𝑑 = (𝑥 +s 𝑞) ↔ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))))
31	19, 27, 30	mp2an 705	. . . . . . . 8 ⊢ (∃𝑞 ∈ ( -s “ (L‘𝑥))𝑑 = (𝑥 +s 𝑞) ↔ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿)))
32	31	abbii 2827	. . . . . . 7 ⊢ {𝑑 ∣ ∃𝑞 ∈ ( -s “ (L‘𝑥))𝑑 = (𝑥 +s 𝑞)} = {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}
33	32	uneq2i 4164	. . . . . 6 ⊢ ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑞 ∈ ( -s “ (L‘𝑥))𝑑 = (𝑥 +s 𝑞)}) = ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))})
34	26, 33	oveq12i 7433	. . . . 5 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑝 ∈ ( -s “ (R‘𝑥))𝑏 = (𝑥 +s 𝑝)}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑞 ∈ ( -s “ (L‘𝑥))𝑑 = (𝑥 +s 𝑞)})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}))
35		fvex 6899	. . . . . . . . . 10 ⊢ (L‘𝑥) ∈ V
36	35	abrexex 7975	. . . . . . . . 9 ⊢ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∈ V
37		fvex 6899	. . . . . . . . . 10 ⊢ (R‘𝑥) ∈ V
38	37	abrexex 7975	. . . . . . . . 9 ⊢ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))} ∈ V
39	36, 38	unex 7762	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) ∈ V
40	39	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) ∈ V)
41		snex 5396	. . . . . . . 8 ⊢ { 0s } ∈ V
42	41	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → { 0s } ∈ V)
43	27	sseli 3904	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ (L‘𝑥) → 𝑥𝐿 ∈ No)
44	43	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → 𝑥𝐿 ∈ No)
45		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → 𝑥 ∈ No)
46	45	negscld 28429	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → ( -s‘𝑥) ∈ No)
47	44, 46	addscld 28372	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑥𝐿 +s ( -s‘𝑥)) ∈ No)
48		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑥𝐿 +s ( -s‘𝑥)) → (𝑎 ∈ No ↔ (𝑥𝐿 +s ( -s‘𝑥)) ∈ No))
49	47, 48	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑎 = (𝑥𝐿 +s ( -s‘𝑥)) → 𝑎 ∈ No))
50	49	rexlimdva 3163	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥)) → 𝑎 ∈ No))
51	50	abssdv 3992	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ⊆ No)
52		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → 𝑥 ∈ No)
53	20	sseli 3904	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑅 ∈ (R‘𝑥) → 𝑥𝑅 ∈ No)
54	53	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → 𝑥𝑅 ∈ No)
55	54	negscld 28429	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → ( -s‘𝑥𝑅) ∈ No)
56	52, 55	addscld 28372	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑥 +s ( -s‘𝑥𝑅)) ∈ No)
57		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑥 +s ( -s‘𝑥𝑅)) → (𝑏 ∈ No ↔ (𝑥 +s ( -s‘𝑥𝑅)) ∈ No))
58	56, 57	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑏 = (𝑥 +s ( -s‘𝑥𝑅)) → 𝑏 ∈ No))
59	58	rexlimdva 3163	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅)) → 𝑏 ∈ No))
60	59	abssdv 3992	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))} ⊆ No)
61	51, 60	unssd 4190	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) ⊆ No)
62		0no 28201	. . . . . . . 8 ⊢ 0s ∈ No
63		snssi 4742	. . . . . . . 8 ⊢ ( 0s ∈ No → { 0s } ⊆ No)
64	62, 63	mp1i 14	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → { 0s } ⊆ No)
65		elun 4152	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) ↔ (𝑝 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∨ 𝑝 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}))
66		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑝 ∈ V
67		eqeq1 2764	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑝 → (𝑎 = (𝑥𝐿 +s ( -s‘𝑥)) ↔ 𝑝 = (𝑥𝐿 +s ( -s‘𝑥))))
68	67	rexbidv 3186	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑝 → (∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥)) ↔ ∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥))))
69	66, 68	elab 3632	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ↔ ∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥)))
70		eqeq1 2764	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑝 → (𝑏 = (𝑥 +s ( -s‘𝑥𝑅)) ↔ 𝑝 = (𝑥 +s ( -s‘𝑥𝑅))))
71	70	rexbidv 3186	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑝 → (∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅)) ↔ ∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅))))
72	66, 71	elab 3632	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))} ↔ ∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅)))
73	69, 72	orbi12i 928	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∨ 𝑝 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) ↔ (∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥)) ∨ ∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅))))
74	65, 73	bitri 278	. . . . . . . . . 10 ⊢ (𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) ↔ (∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥)) ∨ ∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅))))
75		velsn 4596	. . . . . . . . . 10 ⊢ (𝑞 ∈ { 0s } ↔ 𝑞 = 0s )
76	74, 75	anbi12i 640	. . . . . . . . 9 ⊢ ((𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) ∧ 𝑞 ∈ { 0s }) ↔ ((∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥)) ∨ ∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅))) ∧ 𝑞 = 0s ))
77		leftlt 28245	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝐿 ∈ (L‘𝑥) → 𝑥𝐿 <s 𝑥)
78	77	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → 𝑥𝐿 <s 𝑥)
79		ltnegsim 28430	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥𝐿 ∈ No ∧ 𝑥 ∈ No) → (𝑥𝐿 <s 𝑥 → ( -s‘𝑥) <s ( -s‘𝑥𝐿)))
80	44, 45, 79	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑥𝐿 <s 𝑥 → ( -s‘𝑥) <s ( -s‘𝑥𝐿)))
81	78, 80	mpd 16	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → ( -s‘𝑥) <s ( -s‘𝑥𝐿))
82	44	negscld 28429	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → ( -s‘𝑥𝐿) ∈ No)
83	46, 82, 44	ltadds2d 28389	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (( -s‘𝑥) <s ( -s‘𝑥𝐿) ↔ (𝑥𝐿 +s ( -s‘𝑥)) <s (𝑥𝐿 +s ( -s‘𝑥𝐿))))
84	81, 83	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑥𝐿 +s ( -s‘𝑥)) <s (𝑥𝐿 +s ( -s‘𝑥𝐿)))
85		id 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝐿 → 𝑥𝑂 = 𝑥𝐿)
86		fveq2 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝐿 → ( -s‘𝑥𝑂) = ( -s‘𝑥𝐿))
87	85, 86	oveq12d 7439	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 +s ( -s‘𝑥𝑂)) = (𝑥𝐿 +s ( -s‘𝑥𝐿)))
88	87	eqeq1d 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝐿 → ((𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ↔ (𝑥𝐿 +s ( -s‘𝑥𝐿)) = 0s ))
89		simplr 781	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s )
90		elun1 4180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝐿 ∈ (L‘𝑥) → 𝑥𝐿 ∈ ((L‘𝑥) ∪ (R‘𝑥)))
91	90	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → 𝑥𝐿 ∈ ((L‘𝑥) ∪ (R‘𝑥)))
92	88, 89, 91	rspcdva 3577	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑥𝐿 +s ( -s‘𝑥𝐿)) = 0s )
93	84, 92	breqtrd 5130	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑥𝐿 +s ( -s‘𝑥)) <s 0s )
94		breq1 5105	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑥𝐿 +s ( -s‘𝑥)) → (𝑝 <s 0s ↔ (𝑥𝐿 +s ( -s‘𝑥)) <s 0s ))
95	93, 94	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑝 = (𝑥𝐿 +s ( -s‘𝑥)) → 𝑝 <s 0s ))
96	95	rexlimdva 3163	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥)) → 𝑝 <s 0s ))
97	96	imp 412	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥))) → 𝑝 <s 0s )
98		rightgt 28246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑅 ∈ (R‘𝑥) → 𝑥 <s 𝑥𝑅)
99	98	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → 𝑥 <s 𝑥𝑅)
100	52, 54, 55	ltadds1d 28390	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑥 <s 𝑥𝑅 ↔ (𝑥 +s ( -s‘𝑥𝑅)) <s (𝑥𝑅 +s ( -s‘𝑥𝑅))))
101	99, 100	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑥 +s ( -s‘𝑥𝑅)) <s (𝑥𝑅 +s ( -s‘𝑥𝑅)))
102		id 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝑅 → 𝑥𝑂 = 𝑥𝑅)
103		fveq2 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝑅 → ( -s‘𝑥𝑂) = ( -s‘𝑥𝑅))
104	102, 103	oveq12d 7439	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 +s ( -s‘𝑥𝑂)) = (𝑥𝑅 +s ( -s‘𝑥𝑅)))
105	104	eqeq1d 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝑅 → ((𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ↔ (𝑥𝑅 +s ( -s‘𝑥𝑅)) = 0s ))
106		simplr 781	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s )
107		elun2 4181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑅 ∈ (R‘𝑥) → 𝑥𝑅 ∈ ((L‘𝑥) ∪ (R‘𝑥)))
108	107	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → 𝑥𝑅 ∈ ((L‘𝑥) ∪ (R‘𝑥)))
109	105, 106, 108	rspcdva 3577	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑥𝑅 +s ( -s‘𝑥𝑅)) = 0s )
110	101, 109	breqtrd 5130	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑥 +s ( -s‘𝑥𝑅)) <s 0s )
111		breq1 5105	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑥 +s ( -s‘𝑥𝑅)) → (𝑝 <s 0s ↔ (𝑥 +s ( -s‘𝑥𝑅)) <s 0s ))
112	110, 111	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑝 = (𝑥 +s ( -s‘𝑥𝑅)) → 𝑝 <s 0s ))
113	112	rexlimdva 3163	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅)) → 𝑝 <s 0s ))
114	113	imp 412	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅))) → 𝑝 <s 0s )
115	97, 114	jaodan 972	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ (∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥)) ∨ ∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅)))) → 𝑝 <s 0s )
116		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑞 = 0s → (𝑝 <s 𝑞 ↔ 𝑝 <s 0s ))
117	115, 116	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ (∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥)) ∨ ∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅)))) → (𝑞 = 0s → 𝑝 <s 𝑞))
118	117	expimpd 459	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (((∃𝑥𝐿 ∈ (L‘𝑥)𝑝 = (𝑥𝐿 +s ( -s‘𝑥)) ∨ ∃𝑥𝑅 ∈ (R‘𝑥)𝑝 = (𝑥 +s ( -s‘𝑥𝑅))) ∧ 𝑞 = 0s ) → 𝑝 <s 𝑞))
119	76, 118	biimtrid 245	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ((𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) ∧ 𝑞 ∈ { 0s }) → 𝑝 <s 𝑞))
120	119	3impib 1134	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑝 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) ∧ 𝑞 ∈ { 0s }) → 𝑝 <s 𝑞)
121	40, 42, 61, 64, 120	sltsd 28160	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) <<s { 0s })
122	37	abrexex 7975	. . . . . . . . 9 ⊢ {𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∈ V
123	35	abrexex 7975	. . . . . . . . 9 ⊢ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))} ∈ V
124	122, 123	unex 7762	. . . . . . . 8 ⊢ ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}) ∈ V
125	124	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}) ∈ V)
126	52	negscld 28429	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → ( -s‘𝑥) ∈ No)
127	54, 126	addscld 28372	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑥𝑅 +s ( -s‘𝑥)) ∈ No)
128		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑥𝑅 +s ( -s‘𝑥)) → (𝑐 ∈ No ↔ (𝑥𝑅 +s ( -s‘𝑥)) ∈ No))
129	127, 128	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑐 = (𝑥𝑅 +s ( -s‘𝑥)) → 𝑐 ∈ No))
130	129	rexlimdva 3163	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥)) → 𝑐 ∈ No))
131	130	abssdv 3992	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → {𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ⊆ No)
132	45, 82	addscld 28372	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑥 +s ( -s‘𝑥𝐿)) ∈ No)
133		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑥 +s ( -s‘𝑥𝐿)) → (𝑑 ∈ No ↔ (𝑥 +s ( -s‘𝑥𝐿)) ∈ No))
134	132, 133	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑑 = (𝑥 +s ( -s‘𝑥𝐿)) → 𝑑 ∈ No))
135	134	rexlimdva 3163	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿)) → 𝑑 ∈ No))
136	135	abssdv 3992	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))} ⊆ No)
137	131, 136	unssd 4190	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}) ⊆ No)
138		velsn 4596	. . . . . . . . . 10 ⊢ (𝑝 ∈ { 0s } ↔ 𝑝 = 0s )
139		elun 4152	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}) ↔ (𝑞 ∈ {𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∨ 𝑞 ∈ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}))
140		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑞 ∈ V
141		eqeq1 2764	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑞 → (𝑐 = (𝑥𝑅 +s ( -s‘𝑥)) ↔ 𝑞 = (𝑥𝑅 +s ( -s‘𝑥))))
142	141	rexbidv 3186	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑞 → (∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥)) ↔ ∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥))))
143	140, 142	elab 3632	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ {𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ↔ ∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥)))
144		eqeq1 2764	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑞 → (𝑑 = (𝑥 +s ( -s‘𝑥𝐿)) ↔ 𝑞 = (𝑥 +s ( -s‘𝑥𝐿))))
145	144	rexbidv 3186	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑞 → (∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿)) ↔ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿))))
146	140, 145	elab 3632	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))} ↔ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿)))
147	143, 146	orbi12i 928	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ {𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∨ 𝑞 ∈ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}) ↔ (∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) ∨ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿))))
148	139, 147	bitri 278	. . . . . . . . . 10 ⊢ (𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}) ↔ (∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) ∨ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿))))
149	138, 148	anbi12i 640	. . . . . . . . 9 ⊢ ((𝑝 ∈ { 0s } ∧ 𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))})) ↔ (𝑝 = 0s ∧ (∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) ∨ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿)))))
150		ltnegsim 28430	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑅 ∈ No) → (𝑥 <s 𝑥𝑅 → ( -s‘𝑥𝑅) <s ( -s‘𝑥)))
151	52, 54, 150	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑥 <s 𝑥𝑅 → ( -s‘𝑥𝑅) <s ( -s‘𝑥)))
152	99, 151	mpd 16	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → ( -s‘𝑥𝑅) <s ( -s‘𝑥))
153	55, 126, 54	ltadds2d 28389	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (( -s‘𝑥𝑅) <s ( -s‘𝑥) ↔ (𝑥𝑅 +s ( -s‘𝑥𝑅)) <s (𝑥𝑅 +s ( -s‘𝑥))))
154	152, 153	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑥𝑅 +s ( -s‘𝑥𝑅)) <s (𝑥𝑅 +s ( -s‘𝑥)))
155	109, 154	eqbrtrrd 5128	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → 0s <s (𝑥𝑅 +s ( -s‘𝑥)))
156		breq2 5106	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) → ( 0s <s 𝑞 ↔ 0s <s (𝑥𝑅 +s ( -s‘𝑥))))
157	155, 156	syl5ibrcom 250	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) → 0s <s 𝑞))
158	157	rexlimdva 3163	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) → 0s <s 𝑞))
159	158	imp 412	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥))) → 0s <s 𝑞)
160	44, 45, 82	ltadds1d 28390	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑥𝐿 <s 𝑥 ↔ (𝑥𝐿 +s ( -s‘𝑥𝐿)) <s (𝑥 +s ( -s‘𝑥𝐿))))
161	78, 160	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑥𝐿 +s ( -s‘𝑥𝐿)) <s (𝑥 +s ( -s‘𝑥𝐿)))
162	92, 161	eqbrtrrd 5128	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → 0s <s (𝑥 +s ( -s‘𝑥𝐿)))
163		breq2 5106	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = (𝑥 +s ( -s‘𝑥𝐿)) → ( 0s <s 𝑞 ↔ 0s <s (𝑥 +s ( -s‘𝑥𝐿))))
164	162, 163	syl5ibrcom 250	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑞 = (𝑥 +s ( -s‘𝑥𝐿)) → 0s <s 𝑞))
165	164	rexlimdva 3163	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿)) → 0s <s 𝑞))
166	165	imp 412	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿))) → 0s <s 𝑞)
167	159, 166	jaodan 972	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ (∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) ∨ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿)))) → 0s <s 𝑞)
168		breq1 5105	. . . . . . . . . . . 12 ⊢ (𝑝 = 0s → (𝑝 <s 𝑞 ↔ 0s <s 𝑞))
169	167, 168	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ (∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) ∨ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿)))) → (𝑝 = 0s → 𝑝 <s 𝑞))
170	169	ex 418	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ((∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) ∨ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿))) → (𝑝 = 0s → 𝑝 <s 𝑞)))
171	170	impcomd 417	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ((𝑝 = 0s ∧ (∃𝑥𝑅 ∈ (R‘𝑥)𝑞 = (𝑥𝑅 +s ( -s‘𝑥)) ∨ ∃𝑥𝐿 ∈ (L‘𝑥)𝑞 = (𝑥 +s ( -s‘𝑥𝐿)))) → 𝑝 <s 𝑞))
172	149, 171	biimtrid 245	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → ((𝑝 ∈ { 0s } ∧ 𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))})) → 𝑝 <s 𝑞))
173	172	3impib 1134	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) ∧ 𝑝 ∈ { 0s } ∧ 𝑞 ∈ ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))})) → 𝑝 <s 𝑞)
174	42, 125, 64, 137, 173	sltsd 28160	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → { 0s } <<s ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))}))
175	121, 174	cuteq0 28207	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑏 = (𝑥 +s ( -s‘𝑥𝑅))}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑑 = (𝑥 +s ( -s‘𝑥𝐿))})) = 0s )
176	34, 175	eqtrid 2807	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑎 = (𝑥𝐿 +s ( -s‘𝑥))} ∪ {𝑏 ∣ ∃𝑝 ∈ ( -s “ (R‘𝑥))𝑏 = (𝑥 +s 𝑝)}) |s ({𝑐 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑐 = (𝑥𝑅 +s ( -s‘𝑥))} ∪ {𝑑 ∣ ∃𝑞 ∈ ( -s “ (L‘𝑥))𝑑 = (𝑥 +s 𝑞)})) = 0s )
177	18, 176	eqtrd 2795	. . 3 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s ) → (𝑥 +s ( -s‘𝑥)) = 0s )
178	177	ex 418	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(𝑥𝑂 +s ( -s‘𝑥𝑂)) = 0s → (𝑥 +s ( -s‘𝑥)) = 0s ))
179	4, 8, 178	noinds 28337	1 ⊢ (𝐴 ∈ No → (𝐴 +s ( -s‘𝐴)) = 0s )

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  {csn 4584   class class class wbr 5102   “ cima 5653   Fn wfn 6534  ‘cfv 6539  (class class class)co 7421  Nocsur 28003   <s clts 28004   <<s cslts 28149   |s ccuts 28151   0_s c0s 28197  Lcleft 28217  Rcright 28218   +_s cadds 28351   -_scnegs 28411

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-les 28108  df-slts 28150  df-cuts 28152  df-0s 28199  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec 28330  df-norec2 28341  df-adds 28352  df-negs 28413

This theorem is used by:  negsidd  28434  negsex  28435  negnegs  28436  negsdi  28442  subsid  28461  subadds  28462