Theorem negsprop 28085

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		bdayelon 27839	. . . 4 ⊢ ( bday ‘𝐴) ∈ On
2		bdayelon 27839	. . . 4 ⊢ ( bday ‘𝐵) ∈ On
3	1, 2	onun2i 6517	. . 3 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ∈ On
4		risset 3239	. . 3 ⊢ ((( bday ‘𝐴) ∪ ( bday ‘𝐵)) ∈ On ↔ ∃𝑎 ∈ On 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
5	3, 4	mpbi 230	. 2 ⊢ ∃𝑎 ∈ On 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵))
6		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞))))
7	6	imbi1d 341	. . . . . . . 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 3227	. . . . . . 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 6920	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( bday ‘𝑝) = ( bday ‘𝑥))
10	9	uneq1d 4190	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑞)))
11	10	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → (𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞))))
12		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( -us ‘𝑝) = ( -us ‘𝑥))
13	12	eleq1d 2829	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝑥) ∈ No ))
14		breq1 5169	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 <s 𝑞 ↔ 𝑥 <s 𝑞))
15	12	breq2d 5178	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (( -us ‘𝑞) <s ( -us ‘𝑝) ↔ ( -us ‘𝑞) <s ( -us ‘𝑥)))
16	14, 15	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)) ↔ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥))))
17	13, 16	anbi12d 631	. . . . . . . . 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 344	. . . . . . . 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 6920	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → ( bday ‘𝑞) = ( bday ‘𝑦))
20	19	uneq2d 4191	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → (( bday ‘𝑥) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑦)))
21	20	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦))))
22		breq2 5170	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (𝑥 <s 𝑞 ↔ 𝑥 <s 𝑦))
23		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑦 → ( -us ‘𝑞) = ( -us ‘𝑦))
24	23	breq1d 5176	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (( -us ‘𝑞) <s ( -us ‘𝑥) ↔ ( -us ‘𝑦) <s ( -us ‘𝑥)))
25	22, 24	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → ((𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥)) ↔ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))
26	25	anbi2d 629	. . . . . . . . 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 344	. . . . . . . 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 3247	. . . . . . 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 287	. . . . . 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 3331	. . . . . . . . . . 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 3299	. . . . . . . . . . . 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 3239	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ ∃𝑏 ∈ (( bday ‘𝑝) ∪ ( bday ‘𝑞))𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦)))
34	33	imbi1i 349	. . . . . . . . . . . . . 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 278	. . . . . . . . . . . . 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 3134	. . . . . . . . . . . 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 277	. . . . . . . . . . 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 287	. . . . . . . . . 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 484	. . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . 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 28080	. . . . . . . . . . . . 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 1142	. . . . . . . . . . . 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 768	. . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 28084	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 511	. . . . . . . . . . 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 413	. . . . . . . . . 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 253	. . . . . . . . 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 3206	. . . . . . 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 7894	. . . . 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 6920	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( bday ‘𝑝) = ( bday ‘𝐴))
57	56	uneq1d 4190	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝑞)))
58	57	eqeq2d 2751	. . . . . . 7 ⊢ (𝑝 = 𝐴 → (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞))))
59		fveq2 6920	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( -us ‘𝑝) = ( -us ‘𝐴))
60	59	eleq1d 2829	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝐴) ∈ No ))
61		breq1 5169	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (𝑝 <s 𝑞 ↔ 𝐴 <s 𝑞))
62	59	breq2d 5178	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (( -us ‘𝑞) <s ( -us ‘𝑝) ↔ ( -us ‘𝑞) <s ( -us ‘𝐴)))
63	61, 62	imbi12d 344	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → ((𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)) ↔ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴))))
64	60, 63	anbi12d 631	. . . . . . 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 344	. . . . . 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 6920	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → ( bday ‘𝑞) = ( bday ‘𝐵))
67	66	uneq2d 4191	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → (( bday ‘𝐴) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
68	67	eqeq2d 2751	. . . . . . 7 ⊢ (𝑞 = 𝐵 → (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞)) ↔ 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵))))
69		breq2 5170	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (𝐴 <s 𝑞 ↔ 𝐴 <s 𝐵))
70		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑞 = 𝐵 → ( -us ‘𝑞) = ( -us ‘𝐵))
71	70	breq1d 5176	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (( -us ‘𝑞) <s ( -us ‘𝐴) ↔ ( -us ‘𝐵) <s ( -us ‘𝐴)))
72	69, 71	imbi12d 344	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → ((𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴)) ↔ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))
73	72	anbi2d 629	. . . . . . 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 344	. . . . . 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 3646	. . . . 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 3154	. 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 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∪ cun 3974  {csn 4648   class class class wbr 5166   “ cima 5703  Oncon0 6395  ‘cfv 6573   No csur 27702   <s cslt 27703   bday cbday 27704   <<s csslt 27843   L cleft 27902   R cright 27903   -u_s cnegs 28069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-negs 28071

This theorem is referenced by:  negscl  28086  sltnegim  28088  negscut  28089