Theorem negsprop 28421

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 28138	. . . 4 ⊢ ( bday ‘𝐴) ∈ On
2		bdayon 28138	. . . 4 ⊢ ( bday ‘𝐵) ∈ On
3	1, 2	onun2i 6486	. . 3 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ∈ On
4		risset 3238	. . 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 2765	. . . . . . . . 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)) → (( -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 6885	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( bday ‘𝑝) = ( bday ‘𝑥))
10	9	uneq1d 4114	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑞)))
11	10	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → (𝑏 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞))))
12		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → ( -us ‘𝑝) = ( -us ‘𝑥))
13	12	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝑥) ∈ No ))
14		breq1 5106	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 <s 𝑞 ↔ 𝑥 <s 𝑞))
15	12	breq2d 5115	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (( -us ‘𝑞) <s ( -us ‘𝑝) ↔ ( -us ‘𝑞) <s ( -us ‘𝑥)))
16	14, 15	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)) ↔ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥))))
17	13, 16	anbi12d 644	. . . . . . . . 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 347	. . . . . . . 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 6885	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → ( bday ‘𝑞) = ( bday ‘𝑦))
20	19	uneq2d 4115	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → (( bday ‘𝑥) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑦)))
21	20	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦))))
22		breq2 5107	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (𝑥 <s 𝑞 ↔ 𝑥 <s 𝑦))
23		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑦 → ( -us ‘𝑞) = ( -us ‘𝑦))
24	23	breq1d 5113	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (( -us ‘𝑞) <s ( -us ‘𝑥) ↔ ( -us ‘𝑦) <s ( -us ‘𝑥)))
25	22, 24	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → ((𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥)) ↔ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))
26	25	anbi2d 642	. . . . . . . . 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 347	. . . . . . . 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 3245	. . . . . . 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 290	. . . . . 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 3294	. . . . . . . . . . . 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 3190	. . . . . . . . . . . . . 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 3238	. . . . . . . . . . . . . . 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)) → (( -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 281	. . . . . . . . . . . . 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 3138	. . . . . . . . . . . 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 280	. . . . . . . . . . 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 290	. . . . . . . . . 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 490	. . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . 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 28416	. . . . . . . . . . . . 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 1160	. . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 28420	. . . . . . . . . . . . 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 418	. . . . . . . . . . . 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 521	. . . . . . . . . . 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 419	. . . . . . . . . 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 256	. . . . . . . . 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 90	. . . . . . . 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 3204	. . . . . . 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 7868	. . . . 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 6885	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( bday ‘𝑝) = ( bday ‘𝐴))
57	56	uneq1d 4114	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝑞)))
58	57	eqeq2d 2772	. . . . . . 7 ⊢ (𝑝 = 𝐴 → (𝑎 = (( bday ‘𝑝) ∪ ( bday ‘𝑞)) ↔ 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞))))
59		fveq2 6885	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → ( -us ‘𝑝) = ( -us ‘𝐴))
60	59	eleq1d 2846	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝐴) ∈ No ))
61		breq1 5106	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (𝑝 <s 𝑞 ↔ 𝐴 <s 𝑞))
62	59	breq2d 5115	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (( -us ‘𝑞) <s ( -us ‘𝑝) ↔ ( -us ‘𝑞) <s ( -us ‘𝐴)))
63	61, 62	imbi12d 347	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → ((𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)) ↔ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴))))
64	60, 63	anbi12d 644	. . . . . . 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 347	. . . . . 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 6885	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → ( bday ‘𝑞) = ( bday ‘𝐵))
67	66	uneq2d 4115	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → (( bday ‘𝐴) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
68	67	eqeq2d 2772	. . . . . . 7 ⊢ (𝑞 = 𝐵 → (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞)) ↔ 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵))))
69		breq2 5107	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (𝐴 <s 𝑞 ↔ 𝐴 <s 𝐵))
70		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑞 = 𝐵 → ( -us ‘𝑞) = ( -us ‘𝐵))
71	70	breq1d 5113	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (( -us ‘𝑞) <s ( -us ‘𝐴) ↔ ( -us ‘𝐵) <s ( -us ‘𝐴)))
72	69, 71	imbi12d 347	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → ((𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴)) ↔ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))
73	72	anbi2d 642	. . . . . . 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 347	. . . . . 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 3587	. . . . 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 32	. . . 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 87	. . 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 3157	. 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))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ∪ cun 3897  {csn 4584   class class class wbr 5103   “ cima 5654  Oncon0 6362  ‘cfv 6538   No csur 27997   <s clts 27998   bday cbday 27999   <<s cslts 28143   L cleft 28211   R cright 28212   -u_s cnegs 28405

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-0s 28193  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-negs 28407

This theorem is used by:  negscl  28422  ltnegsim  28424  negcut  28425