Theorem negsprop 28430

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 28147	. . . 4 ⊢ (bday‘𝐴) ∈ On
2		bdayon 28147	. . . 4 ⊢ (bday‘𝐵) ∈ On
3	1, 2	onun2i 6489	. . 3 ⊢ ((bday‘𝐴) ∪ (bday‘𝐵)) ∈ On
4		risset 3237	. . 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 2764	. . . . . . . . 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 3226	. . . . . . 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 6890	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (bday‘𝑝) = (bday‘𝑥))
10	9	uneq1d 4165	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((bday‘𝑝) ∪ (bday‘𝑞)) = ((bday‘𝑥) ∪ (bday‘𝑞)))
11	10	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → (𝑏 = ((bday‘𝑝) ∪ (bday‘𝑞)) ↔ 𝑏 = ((bday‘𝑥) ∪ (bday‘𝑞))))
12		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( -s‘𝑝) = ( -s‘𝑥))
13	12	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( -s‘𝑝) ∈ No ↔ ( -s‘𝑥) ∈ No))
14		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 <s 𝑞 ↔ 𝑥 <s 𝑞))
15	12	breq2d 5114	. . . . . . . . . . 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 6890	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (bday‘𝑞) = (bday‘𝑦))
20	19	uneq2d 4166	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → ((bday‘𝑥) ∪ (bday‘𝑞)) = ((bday‘𝑥) ∪ (bday‘𝑦)))
21	20	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑏 = ((bday‘𝑥) ∪ (bday‘𝑞)) ↔ 𝑏 = ((bday‘𝑥) ∪ (bday‘𝑦))))
22		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (𝑥 <s 𝑞 ↔ 𝑥 <s 𝑦))
23		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑦 → ( -s‘𝑞) = ( -s‘𝑦))
24	23	breq1d 5112	. . . . . . . . . . 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 3244	. . . . . . 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 3316	. . . . . . . . . . 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 3293	. . . . . . . . . . . 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 3189	. . . . . . . . . . . . . 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 3237	. . . . . . . . . . . . . . 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 3137	. . . . . . . . . . . 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 28425	. . . . . . . . . . . . 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 28429	. . . . . . . . . . . . 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 3203	. . . . . . 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 7872	. . . . 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 6890	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (bday‘𝑝) = (bday‘𝐴))
57	56	uneq1d 4165	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → ((bday‘𝑝) ∪ (bday‘𝑞)) = ((bday‘𝐴) ∪ (bday‘𝑞)))
58	57	eqeq2d 2771	. . . . . . 7 ⊢ (𝑝 = 𝐴 → (𝑎 = ((bday‘𝑝) ∪ (bday‘𝑞)) ↔ 𝑎 = ((bday‘𝐴) ∪ (bday‘𝑞))))
59		fveq2 6890	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( -s‘𝑝) = ( -s‘𝐴))
60	59	eleq1d 2845	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( -s‘𝑝) ∈ No ↔ ( -s‘𝐴) ∈ No))
61		breq1 5105	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (𝑝 <s 𝑞 ↔ 𝐴 <s 𝑞))
62	59	breq2d 5114	. . . . . . . . 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 6890	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (bday‘𝑞) = (bday‘𝐵))
67	66	uneq2d 4166	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → ((bday‘𝐴) ∪ (bday‘𝑞)) = ((bday‘𝐴) ∪ (bday‘𝐵)))
68	67	eqeq2d 2771	. . . . . . 7 ⊢ (𝑞 = 𝐵 → (𝑎 = ((bday‘𝐴) ∪ (bday‘𝑞)) ↔ 𝑎 = ((bday‘𝐴) ∪ (bday‘𝐵))))
69		breq2 5106	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (𝐴 <s 𝑞 ↔ 𝐴 <s 𝐵))
70		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑞 = 𝐵 → ( -s‘𝑞) = ( -s‘𝐵))
71	70	breq1d 5112	. . . . . . . . 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 3586	. . . . 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 3156	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ∪ cun 4148  {csn 4584   class class class wbr 5102   “ cima 5654  Oncon0 6365  ‘cfv 6543  Nocsur 28006   <s clts 28007  bdaycbday 28008   <<s cslts 28152  Lcleft 28220  Rcright 28221   -_scnegs 28414

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 5327  ax-pr 5391  ax-un 7755

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 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-1o 8484  df-2o 8485  df-no 28009  df-lts 28010  df-bday 28011  df-slts 28153  df-cuts 28155  df-0s 28202  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec 28333  df-negs 28416

This theorem is used by:  negscl  28431  ltnegsim  28433  negcut  28434