Theorem negsprop 28125

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

Assertion

Ref	Expression
negsprop	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘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 27842	. . . 4 ⊢ ( bday ‘𝐴) ∈ On
2		bdayon 27842	. . . 4 ⊢ ( bday ‘𝐵) ∈ On
3	1, 2	onun2i 6469	. . 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 232	. 2 ⊢ ∃𝑎 ∈ On 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵))
6		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞))))
7	6	imbi1d 343	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ (𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
8	7	2ralbidv 3226	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ ∀𝑝 ∈ No ∀𝑞 ∈ No (𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
9		fveq2 6867	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( bday ‘𝑝) = ( bday ‘𝑥))
10	9	uneq1d 4120	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑞)))
11	10	eqeq2d 2773	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → (𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞))))
12		fveq2 6867	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( -us ‘𝑝) = ( -us ‘𝑥))
13	12	eleq1d 2847	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝑥) ∈ No ))
14		breq1 5103	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 <s 𝑞 ↔ 𝑥 <s 𝑞))
15	12	breq2d 5112	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (( -us ‘𝑞) <s ( -us ‘𝑝) ↔ ( -us ‘𝑞) <s ( -us ‘𝑥)))
16	14, 15	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)) ↔ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥))))
17	13, 16	anbi12d 641	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → ((( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))) ↔ (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥)))))
18	11, 17	imbi12d 346	. . . . . . . 8 ⊢ (𝑝 = 𝑥 → ((𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥))))))
19		fveq2 6867	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → ( bday ‘𝑞) = ( bday ‘𝑦))
20	19	uneq2d 4121	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → (( bday ‘𝑥) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑦)))
21	20	eqeq2d 2773	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦))))
22		breq2 5104	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (𝑥 <s 𝑞 ↔ 𝑥 <s 𝑦))
23		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑦 → ( -us ‘𝑞) = ( -us ‘𝑦))
24	23	breq1d 5110	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (( -us ‘𝑞) <s ( -us ‘𝑥) ↔ ( -us ‘𝑦) <s ( -us ‘𝑥)))
25	22, 24	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → ((𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥)) ↔ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))
26	25	anbi2d 639	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → ((( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥))) ↔ (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
27	21, 26	imbi12d 346	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ((𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥)))) ↔ (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))))
28	18, 27	cbvral2vw 3244	. . . . . . 7 ⊢ (∀𝑝 ∈ No ∀𝑞 ∈ No (𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
29	8, 28	bitrdi 289	. . . . . 6 ⊢ (𝑎 = 𝑏 → (∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))))
30		raleq 3317	. . . . . . . . . . 11 ⊢ (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))))
31		ralrot3 3293	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))(𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
32		r19.23v 3189	. . . . . . . . . . . . . 14 ⊢ (∀𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))(𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ (∃𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘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 351	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ (∃𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
35	32, 34	bitr4i 280	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))(𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
36	35	2ralbii 3137	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))(𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
37	31, 36	bitr3i 279	. . . . . . . . . . 11 ⊢ (∀𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
38	30, 37	bitrdi 289	. . . . . . . . . 10 ⊢ (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))))
39		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
40		simpll 776	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → 𝑝 ∈ No )
41	39, 40	negsproplem3 28120	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → (( -us ‘𝑝) ∈ No ∧ ( -us “ ( R ‘𝑝)) <<s {( -us ‘𝑝)} ∧ {( -us ‘𝑝)} <<s ( -us “ ( L ‘𝑝))))
42	41	simp1d 1155	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → ( -us ‘𝑝) ∈ No )
43		simplr 778	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
44		simplll 784	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → 𝑝 ∈ No )
45		simpllr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → 𝑞 ∈ No )
46		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → 𝑝 <s 𝑞)
47	43, 44, 45, 46	negsproplem7 28124	. . . . . . . . . . . . 13 ⊢ ((((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → ( -us ‘𝑞) <s ( -us ‘𝑝))
48	47	ex 416	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))
49	42, 48	jca 519	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ No ∧ 𝑞 ∈ No ) ∧ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))
50	49	expcom 417	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ((𝑝 ∈ No ∧ 𝑞 ∈ No ) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))))
51	38, 50	biimtrdi 255	. . . . . . . . 9 ⊢ (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ((𝑝 ∈ No ∧ 𝑞 ∈ No ) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
52	51	com3l 89	. . . . . . . 8 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ((𝑝 ∈ No ∧ 𝑞 ∈ No ) → (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
53	52	ralrimivv 3203	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))))
54	53	a1i 11	. . . . . 6 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑥 ∈ No ∀𝑦 ∈ No (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
55	29, 54	tfis2 7837	. . . . 5 ⊢ (𝑎 ∈ On → ∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))))
56		fveq2 6867	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( bday ‘𝑝) = ( bday ‘𝐴))
57	56	uneq1d 4120	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝑞)))
58	57	eqeq2d 2773	. . . . . . 7 ⊢ (𝑝 = 𝐴 → (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞))))
59		fveq2 6867	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( -us ‘𝑝) = ( -us ‘𝐴))
60	59	eleq1d 2847	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝐴) ∈ No ))
61		breq1 5103	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (𝑝 <s 𝑞 ↔ 𝐴 <s 𝑞))
62	59	breq2d 5112	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (( -us ‘𝑞) <s ( -us ‘𝑝) ↔ ( -us ‘𝑞) <s ( -us ‘𝐴)))
63	61, 62	imbi12d 346	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → ((𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)) ↔ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴))))
64	60, 63	anbi12d 641	. . . . . . 7 ⊢ (𝑝 = 𝐴 → ((( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))) ↔ (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴)))))
65	58, 64	imbi12d 346	. . . . . 6 ⊢ (𝑝 = 𝐴 → ((𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴))))))
66		fveq2 6867	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → ( bday ‘𝑞) = ( bday ‘𝐵))
67	66	uneq2d 4121	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → (( bday ‘𝐴) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
68	67	eqeq2d 2773	. . . . . . 7 ⊢ (𝑞 = 𝐵 → (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞)) ↔ 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵))))
69		breq2 5104	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (𝐴 <s 𝑞 ↔ 𝐴 <s 𝐵))
70		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑞 = 𝐵 → ( -us ‘𝑞) = ( -us ‘𝐵))
71	70	breq1d 5110	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (( -us ‘𝑞) <s ( -us ‘𝐴) ↔ ( -us ‘𝐵) <s ( -us ‘𝐴)))
72	69, 71	imbi12d 346	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → ((𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴)) ↔ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))
73	72	anbi2d 639	. . . . . . 7 ⊢ (𝑞 = 𝐵 → ((( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴))) ↔ (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴)))))
74	68, 73	imbi12d 346	. . . . . 6 ⊢ (𝑞 = 𝐵 → ((𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴)))) ↔ (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))))
75	65, 74	rspc2v 3592	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∀𝑝 ∈ No ∀𝑞 ∈ No (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) → (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))))
76	55, 75	syl5com 31	. . . 4 ⊢ (𝑎 ∈ On → ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))))
77	76	com23 86	. . 3 ⊢ (𝑎 ∈ On → (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))))
78	77	rexlimiv 3156	. 2 ⊢ (∃𝑎 ∈ On 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴)))))
79	5, 78	ax-mp 5	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ∪ cun 3902  {csn 4582   class class class wbr 5100   “ cima 5650  Oncon0 6346  ‘cfv 6521   No csur 27701   <s clts 27702   bday cbday 27703   <<s cslts 27847   L cleft 27915   R cright 27916   -u_s cnegs 28109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-1o 8437  df-2o 8438  df-no 27704  df-lts 27705  df-bday 27706  df-slts 27848  df-cuts 27850  df-0s 27897  df-made 27917  df-old 27918  df-left 27920  df-right 27921  df-norec 28028  df-negs 28111

This theorem is referenced by:  negscl  28126  ltnegsim  28128  negcut  28129