Theorem negsprop 28478

Description: Show closure and ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)

Assertion

Ref	Expression
negsprop	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴))))

Proof of Theorem negsprop

Dummy variables 𝑎 𝑏 𝑝 𝑞 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdayon 28195	. . . 4 ⊢ (bday‘𝐴) ∈ On
2		bdayon 28195	. . . 4 ⊢ (bday‘𝐵) ∈ On
3	1, 2	onun2i 6512	. . 3 ⊢ ((bday‘𝐴) ∪ (bday‘𝐵)) ∈ On
4		risset 3241	. . 3 ⊢ (((bday‘𝐴) ∪ (bday‘𝐵)) ∈ On ↔ ∃𝑎 ∈ On 𝑎 = ((bday‘𝐴) ∪ (bday‘𝐵)))
5	3, 4	mpbi 233	. 2 ⊢ ∃𝑎 ∈ On 𝑎 = ((bday‘𝐴) ∪ (bday‘𝐵))
6		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) ↔ 𝑏 = ((bday‘𝑝) ∪ (bday‘𝑞))))
7	6	imbi1d 344	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))) ↔ (𝑏 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝))))))
8	7	2ralbidv 3230	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))) ↔ ∀𝑝 ∈ No ∀𝑞 ∈ No (𝑏 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝))))))
9		fveq2 6916	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (bday‘𝑝) = (bday‘𝑥))
10	9	uneq1d 4169	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((bday‘𝑝) ∪ (bday‘𝑞)) = ((bday‘𝑥) ∪ (bday‘𝑞)))
11	10	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → (𝑏 = ((bday‘𝑝) ∪ (bday‘𝑞)) ↔ 𝑏 = ((bday‘𝑥) ∪ (bday‘𝑞))))
12		fveq2 6916	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( -s‘𝑝) = ( -s‘𝑥))
13	12	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( -s‘𝑝) ∈ No ↔ ( -s‘𝑥) ∈ No))
14		breq1 5111	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 <s 𝑞 ↔ 𝑥 <s 𝑞))
15	12	breq2d 5120	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (( -s‘𝑞) <s ( -s‘𝑝) ↔ ( -s‘𝑞) <s ( -s‘𝑥)))
16	14, 15	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)) ↔ (𝑥 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑥))))
17	13, 16	anbi12d 644	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → ((( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝))) ↔ (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑥)))))
18	11, 17	imbi12d 347	. . . . . . . 8 ⊢ (𝑝 = 𝑥 → ((𝑏 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))) ↔ (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑥))))))
19		fveq2 6916	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (bday‘𝑞) = (bday‘𝑦))
20	19	uneq2d 4170	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → ((bday‘𝑥) ∪ (bday‘𝑞)) = ((bday‘𝑥) ∪ (bday‘𝑦)))
21	20	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑞)) ↔ 𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦))))
22		breq2 5112	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (𝑥 <s 𝑞 ↔ 𝑥 <s 𝑦))
23		fveq2 6916	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑦 → ( -s‘𝑞) = ( -s‘𝑦))
24	23	breq1d 5118	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (( -s‘𝑞) <s ( -s‘𝑥) ↔ ( -s‘𝑦) <s ( -s‘𝑥)))
25	22, 24	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → ((𝑥 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑥)) ↔ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))
26	25	anbi2d 642	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → ((( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑥))) ↔ (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
27	21, 26	imbi12d 347	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ((𝑏 = ((bday‘𝑥) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑥)))) ↔ (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))))
28	18, 27	cbvral2vw 3248	. . . . . . 7 ⊢ (∀𝑝 ∈ No ∀𝑞 ∈ No (𝑏 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
29	8, 28	bitrdi 290	. . . . . 6 ⊢ (𝑎 = 𝑏 → (∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))))
30		raleq 3320	. . . . . . . . . . 11 ⊢ (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ ∀𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))))
31		ralrot3 3297	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))(𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ ∀𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
32		r19.23v 3193	. . . . . . . . . . . . . 14 ⊢ (∀𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))(𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ (∃𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
33		risset 3241	. . . . . . . . . . . . . . 15 ⊢ (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) ↔ ∃𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)))
34	33	imbi1i 352	. . . . . . . . . . . . . 14 ⊢ ((((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ (∃𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
35	32, 34	bitr4i 281	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))(𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
36	35	2ralbii 3141	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))(𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
37	31, 36	bitr3i 280	. . . . . . . . . . 11 ⊢ (∀𝑏 ∈ ((bday‘𝑝) ∪ (bday‘𝑞))∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
38	30, 37	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))))
39		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
40		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) → 𝑝 ∈ No)
41	39, 40	negsproplem3 28473	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) → (( -s‘𝑝) ∈ No ∧ ( -s “ (R‘𝑝)) <<s {( -s‘𝑝)} ∧ {( -s‘𝑝)} <<s ( -s “ (L‘𝑝))))
42	41	simp1d 1160	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) → ( -s‘𝑝) ∈ No)
43		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) ∧ 𝑝 <s 𝑞) → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
44		simplll 787	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) ∧ 𝑝 <s 𝑞) → 𝑝 ∈ No)
45		simpllr 788	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) ∧ 𝑝 <s 𝑞) → 𝑞 ∈ No)
46		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) ∧ 𝑝 <s 𝑞) → 𝑝 <s 𝑞)
47	43, 44, 45, 46	negsproplem7 28477	. . . . . . . . . . . . 13 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) ∧ 𝑝 <s 𝑞) → ( -s‘𝑞) <s ( -s‘𝑝))
48	47	ex 418	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) → (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))
49	42, 48	jca 521	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))))) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝))))
50	49	expcom 419	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) → ((𝑝 ∈ No ∧ 𝑞 ∈ No) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))))
51	38, 50	biimtrdi 256	. . . . . . . . 9 ⊢ (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) → ((𝑝 ∈ No ∧ 𝑞 ∈ No) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝))))))
52	51	com3l 90	. . . . . . . 8 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) → ((𝑝 ∈ No ∧ 𝑞 ∈ No) → (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝))))))
53	52	ralrimivv 3207	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) → ∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))))
54	53	a1i 11	. . . . . 6 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) → ∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝))))))
55	29, 54	tfis2 7902	. . . . 5 ⊢ (𝑎 ∈ On → ∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))))
56		fveq2 6916	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (bday‘𝑝) = (bday‘𝐴))
57	56	uneq1d 4169	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → ((bday‘𝑝) ∪ (bday‘𝑞)) = ((bday‘𝐴) ∪ (bday‘𝑞)))
58	57	eqeq2d 2774	. . . . . . 7 ⊢ (𝑝 = 𝐴 → (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) ↔ 𝑎 = ((bday‘𝐴) ∪ (bday‘𝑞))))
59		fveq2 6916	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( -s‘𝑝) = ( -s‘𝐴))
60	59	eleq1d 2849	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( -s‘𝑝) ∈ No ↔ ( -s‘𝐴) ∈ No))
61		breq1 5111	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (𝑝 <s 𝑞 ↔ 𝐴 <s 𝑞))
62	59	breq2d 5120	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (( -s‘𝑞) <s ( -s‘𝑝) ↔ ( -s‘𝑞) <s ( -s‘𝐴)))
63	61, 62	imbi12d 347	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → ((𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)) ↔ (𝐴 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝐴))))
64	60, 63	anbi12d 644	. . . . . . 7 ⊢ (𝑝 = 𝐴 → ((( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝))) ↔ (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝐴)))))
65	58, 64	imbi12d 347	. . . . . 6 ⊢ (𝑝 = 𝐴 → ((𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))) ↔ (𝑎 = ((bday‘𝐴) ∪ (bday‘𝑞)) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝐴))))))
66		fveq2 6916	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (bday‘𝑞) = (bday‘𝐵))
67	66	uneq2d 4170	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → ((bday‘𝐴) ∪ (bday‘𝑞)) = ((bday‘𝐴) ∪ (bday‘𝐵)))
68	67	eqeq2d 2774	. . . . . . 7 ⊢ (𝑞 = 𝐵 → (𝑎 = ((bday‘𝐴) ∪ (bday‘𝑞)) ↔ 𝑎 = ((bday‘𝐴) ∪ (bday‘𝐵))))
69		breq2 5112	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (𝐴 <s 𝑞 ↔ 𝐴 <s 𝐵))
70		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑞 = 𝐵 → ( -s‘𝑞) = ( -s‘𝐵))
71	70	breq1d 5118	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (( -s‘𝑞) <s ( -s‘𝐴) ↔ ( -s‘𝐵) <s ( -s‘𝐴)))
72	69, 71	imbi12d 347	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → ((𝐴 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝐴)) ↔ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴))))
73	72	anbi2d 642	. . . . . . 7 ⊢ (𝑞 = 𝐵 → ((( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝐴))) ↔ (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴)))))
74	68, 73	imbi12d 347	. . . . . 6 ⊢ (𝑞 = 𝐵 → ((𝑎 = ((bday‘𝐴) ∪ (bday‘𝑞)) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝐴)))) ↔ (𝑎 = ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴))))))
75	65, 74	rspc2v 3590	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) → (( -s‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -s‘𝑞) <s ( -s‘𝑝)))) → (𝑎 = ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴))))))
76	55, 75	syl5com 32	. . . 4 ⊢ (𝑎 ∈ On → ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑎 = ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴))))))
77	76	com23 87	. . 3 ⊢ (𝑎 ∈ On → (𝑎 = ((bday‘𝐴) ∪ (bday‘𝐵)) → ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴))))))
78	77	rexlimiv 3160	. 2 ⊢ (∃𝑎 ∈ On 𝑎 = ((bday‘𝐴) ∪ (bday‘𝐵)) → ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴)))))
79	5, 78	ax-mp 5	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ∪ cun 4152  {csn 4588   class class class wbr 5108   “ cima 5661  Oncon0 6388  ‘cfv 6566  Nocsur 28054   <s clts 28055  bdaycbday 28056   <<s cslts 28200  Lcleft 28268  Rcright 28269   -_scnegs 28462

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-0s 28250  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-negs 28464

This theorem is used by:  negscl  28479  ltnegsim  28481  negcut  28482