Theorem negsprop 28067

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 27821	. . . 4 ⊢ ( bday ‘𝐴) ∈ On
2		bdayelon 27821	. . . 4 ⊢ ( bday ‘𝐵) ∈ On
3	1, 2	onun2i 6506	. . 3 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ∈ On
4		risset 3233	. . 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 2741	. . . . . . . . 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 3221	. . . . . . 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 6906	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( bday ‘𝑝) = ( bday ‘𝑥))
10	9	uneq1d 4167	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑞)))
11	10	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → (𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞))))
12		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( -us ‘𝑝) = ( -us ‘𝑥))
13	12	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝑥) ∈ No ))
14		breq1 5146	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 <s 𝑞 ↔ 𝑥 <s 𝑞))
15	12	breq2d 5155	. . . . . . . . . . 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 632	. . . . . . . . 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 6906	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → ( bday ‘𝑞) = ( bday ‘𝑦))
20	19	uneq2d 4168	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → (( bday ‘𝑥) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑦)))
21	20	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦))))
22		breq2 5147	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (𝑥 <s 𝑞 ↔ 𝑥 <s 𝑦))
23		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑦 → ( -us ‘𝑞) = ( -us ‘𝑦))
24	23	breq1d 5153	. . . . . . . . . . 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 630	. . . . . . . . 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 3241	. . . . . . 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 3323	. . . . . . . . . . 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 3183	. . . . . . . . . . . . . 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 3233	. . . . . . . . . . . . . . 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 3128	. . . . . . . . . . . 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 767	. . . . . . . . . . . . . 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 28062	. . . . . . . . . . . . 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 1143	. . . . . . . . . . . 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 769	. . . . . . . . . . . . . 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 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) → 𝑝 ∈ No )
45		simpllr 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 <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 28066	. . . . . . . . . . . . 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 3200	. . . . . . 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 7878	. . . . 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 6906	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( bday ‘𝑝) = ( bday ‘𝐴))
57	56	uneq1d 4167	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝑞)))
58	57	eqeq2d 2748	. . . . . . 7 ⊢ (𝑝 = 𝐴 → (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞))))
59		fveq2 6906	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( -us ‘𝑝) = ( -us ‘𝐴))
60	59	eleq1d 2826	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝐴) ∈ No ))
61		breq1 5146	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (𝑝 <s 𝑞 ↔ 𝐴 <s 𝑞))
62	59	breq2d 5155	. . . . . . . . 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 632	. . . . . . 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 6906	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → ( bday ‘𝑞) = ( bday ‘𝐵))
67	66	uneq2d 4168	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → (( bday ‘𝐴) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
68	67	eqeq2d 2748	. . . . . . 7 ⊢ (𝑞 = 𝐵 → (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞)) ↔ 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵))))
69		breq2 5147	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (𝐴 <s 𝑞 ↔ 𝐴 <s 𝐵))
70		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑞 = 𝐵 → ( -us ‘𝑞) = ( -us ‘𝐵))
71	70	breq1d 5153	. . . . . . . . 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 630	. . . . . . 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 3633	. . . . 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 3148	. 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 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∪ cun 3949  {csn 4626   class class class wbr 5143   “ cima 5688  Oncon0 6384  ‘cfv 6561   No csur 27684   <s cslt 27685   bday cbday 27686   <<s csslt 27825   L cleft 27884   R cright 27885   -u_s cnegs 28051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-negs 28053

This theorem is referenced by:  negscl  28068  sltnegim  28070  negscut  28071