Theorem negsprop 28279

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 27996	. . . 4 ⊢ ( bday ‘𝐴) ∈ On
2		bdayon 27996	. . . 4 ⊢ ( bday ‘𝐵) ∈ On
3	1, 2	onun2i 6488	. . 3 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ∈ On
4		risset 3242	. . 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 2769	. . . . . . . . 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 3231	. . . . . . 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 4121	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑞)))
11	10	eqeq2d 2776	. . . . . . . . 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 2850	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝑥) ∈ No ))
14		breq1 5114	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → (𝑝 <s 𝑞 ↔ 𝑥 <s 𝑞))
15	12	breq2d 5123	. . . . . . . . . . 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 4122	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → (( bday ‘𝑥) ∪ ( bday ‘𝑞)) = (( bday ‘𝑥) ∪ ( bday ‘𝑦)))
21	20	eqeq2d 2776	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑞)) ↔ 𝑏 = (( bday ‘𝑥) ∪ ( bday ‘𝑦))))
22		breq2 5115	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑦 → (𝑥 <s 𝑞 ↔ 𝑥 <s 𝑦))
23		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑦 → ( -us ‘𝑞) = ( -us ‘𝑦))
24	23	breq1d 5121	. . . . . . . . . . 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 3249	. . . . . . 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 3322	. . . . . . . . . . 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 3298	. . . . . . . . . . . 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 3194	. . . . . . . . . . . . . 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 3242	. . . . . . . . . . . . . . 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 3142	. . . . . . . . . . . 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 28274	. . . . . . . . . . . . 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 28278	. . . . . . . . . . . . 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 3208	. . . . . . 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 7859	. . . . 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 4121	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( bday ‘𝑝) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝑞)))
58	57	eqeq2d 2776	. . . . . . 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 2850	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝐴) ∈ No ))
61		breq1 5114	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → (𝑝 <s 𝑞 ↔ 𝐴 <s 𝑞))
62	59	breq2d 5123	. . . . . . . . 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 4122	. . . . . . . 8 ⊢ (𝑞 = 𝐵 → (( bday ‘𝐴) ∪ ( bday ‘𝑞)) = (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
68	67	eqeq2d 2776	. . . . . . 7 ⊢ (𝑞 = 𝐵 → (𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝑞)) ↔ 𝑎 = (( bday ‘𝐴) ∪ ( bday ‘𝐵))))
69		breq2 5115	. . . . . . . . 9 ⊢ (𝑞 = 𝐵 → (𝐴 <s 𝑞 ↔ 𝐴 <s 𝐵))
70		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑞 = 𝐵 → ( -us ‘𝑞) = ( -us ‘𝐵))
71	70	breq1d 5121	. . . . . . . . 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 3594	. . . . 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 3161	. 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 2146  ∀wral 3081  ∃wrex 3091   ∪ cun 3904  {csn 4591   class class class wbr 5111   “ cima 5666  Oncon0 6364  ‘cfv 6540   No csur 27855   <s clts 27856   bday cbday 27857   <<s cslts 28001   L cleft 28069   R cright 28070   -u_s cnegs 28263

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-1o 8459  df-2o 8460  df-no 27858  df-lts 27859  df-bday 27860  df-slts 28002  df-cuts 28004  df-0s 28051  df-made 28071  df-old 28072  df-left 28074  df-right 28075  df-norec 28182  df-negs 28265

This theorem is used by:  negscl  28280  ltnegsim  28282  negcut  28283