Theorem nocvxminlem 27723

Description: Lemma for nocvxmin 27724. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)

Assertion

Ref	Expression
nocvxminlem	⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴))) → ¬ 𝑋 <s 𝑌))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑦,𝑌,𝑧

Allowed substitution hint:   𝑌(𝑥)

Proof of Theorem nocvxminlem

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5096	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → (𝑥 <s 𝑧 ↔ 𝑋 <s 𝑧))
2	1	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦)))
3	2	imbi1d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) ↔ ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)))
4	3	ralbidv 3155	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) ↔ ∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)))
5		breq2 5097	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑧 <s 𝑦 ↔ 𝑧 <s 𝑌))
6	5	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌)))
7	6	imbi1d 341	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → (((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) ↔ ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴)))
8	7	ralbidv 3155	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) ↔ ∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴)))
9	4, 8	rspc2v 3583	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) → ∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴)))
10		breq2 5097	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑤))
11		breq1 5096	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑧 <s 𝑌 ↔ 𝑤 <s 𝑌))
12	10, 11	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) ↔ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
13		eleq1w 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
14	12, 13	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) ↔ ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → 𝑤 ∈ 𝐴)))
15	14	rspcv 3568	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ No → (∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → 𝑤 ∈ 𝐴)))
16		bdaydm 27719	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom bday = No
17	16	sseq2i 3959	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
18		bdayfun 27717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Fun bday
19		funfvima2 7171	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday ) → (𝑤 ∈ 𝐴 → ( bday ‘𝑤) ∈ ( bday “ 𝐴)))
20	18, 19	mpan 690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ dom bday → (𝑤 ∈ 𝐴 → ( bday ‘𝑤) ∈ ( bday “ 𝐴)))
21	17, 20	sylbir 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → ( bday ‘𝑤) ∈ ( bday “ 𝐴)))
22	21	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ( bday ‘𝑤) ∈ ( bday “ 𝐴))
23		intss1 4913	. . . . . . . . . . . . . . . . . . 19 ⊢ (( bday ‘𝑤) ∈ ( bday “ 𝐴) → ∩ ( bday “ 𝐴) ⊆ ( bday ‘𝑤))
24	22, 23	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ∩ ( bday “ 𝐴) ⊆ ( bday ‘𝑤))
25		imassrn 6025	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday “ 𝐴) ⊆ ran bday
26		bdayrn 27720	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran bday = On
27	25, 26	sseqtri 3978	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( bday “ 𝐴) ⊆ On
28	22	ne0d 4291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ( bday “ 𝐴) ≠ ∅)
29		oninton 7734	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((( bday “ 𝐴) ⊆ On ∧ ( bday “ 𝐴) ≠ ∅) → ∩ ( bday “ 𝐴) ∈ On)
30	27, 28, 29	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ∩ ( bday “ 𝐴) ∈ On)
31		bdayelon 27721	. . . . . . . . . . . . . . . . . . 19 ⊢ ( bday ‘𝑤) ∈ On
32		ontri1 6346	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∩ ( bday “ 𝐴) ∈ On ∧ ( bday ‘𝑤) ∈ On) → (∩ ( bday “ 𝐴) ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ ∩ ( bday “ 𝐴)))
33	30, 31, 32	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → (∩ ( bday “ 𝐴) ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ ∩ ( bday “ 𝐴)))
34	24, 33	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ¬ ( bday ‘𝑤) ∈ ∩ ( bday “ 𝐴))
35	34	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → ¬ ( bday ‘𝑤) ∈ ∩ ( bday “ 𝐴)))
36		eleq2 2820	. . . . . . . . . . . . . . . . . 18 ⊢ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) → (( bday ‘𝑤) ∈ ( bday ‘𝑋) ↔ ( bday ‘𝑤) ∈ ∩ ( bday “ 𝐴)))
37	36	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) → (¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋) ↔ ¬ ( bday ‘𝑤) ∈ ∩ ( bday “ 𝐴)))
38	37	biimprcd 250	. . . . . . . . . . . . . . . 16 ⊢ (¬ ( bday ‘𝑤) ∈ ∩ ( bday “ 𝐴) → (( bday ‘𝑋) = ∩ ( bday “ 𝐴) → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋)))
39	35, 38	syl6 35	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → (( bday ‘𝑋) = ∩ ( bday “ 𝐴) → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋))))
40	39	com3l 89	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝐴 → (( bday ‘𝑋) = ∩ ( bday “ 𝐴) → (𝐴 ⊆ No → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋))))
41	40	adantrd 491	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝐴 → ((( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)) → (𝐴 ⊆ No → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋))))
42	15, 41	syl8 76	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ No → (∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → ((( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)) → (𝐴 ⊆ No → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋))))))
43	42	com35 98	. . . . . . . . . . 11 ⊢ (𝑤 ∈ No → (∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) → (𝐴 ⊆ No → ((( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)) → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋))))))
44	43	com4l 92	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) → (𝐴 ⊆ No → ((( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)) → (𝑤 ∈ No → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋))))))
45	9, 44	syl6 35	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) → (𝐴 ⊆ No → ((( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)) → (𝑤 ∈ No → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋)))))))
46	45	com3l 89	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) → (𝐴 ⊆ No → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)) → (𝑤 ∈ No → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋)))))))
47	46	impcom 407	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)) → (𝑤 ∈ No → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋))))))
48	47	imp42 426	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)))) ∧ 𝑤 ∈ No ) → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋)))
49	48	con2d 134	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)))) ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) ∈ ( bday ‘𝑋) → ¬ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
50		3anass 1094	. . . . . . 7 ⊢ ((( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) ↔ (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
51	50	notbii 320	. . . . . 6 ⊢ (¬ (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) ↔ ¬ (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
52		imnan 399	. . . . . 6 ⊢ ((( bday ‘𝑤) ∈ ( bday ‘𝑋) → ¬ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)) ↔ ¬ (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
53	51, 52	bitr4i 278	. . . . 5 ⊢ (¬ (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) ↔ (( bday ‘𝑤) ∈ ( bday ‘𝑋) → ¬ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
54	49, 53	sylibr 234	. . . 4 ⊢ ((((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)))) ∧ 𝑤 ∈ No ) → ¬ (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌))
55	54	nrexdv 3127	. . 3 ⊢ (((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)))) → ¬ ∃𝑤 ∈ No (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌))
56		ssel 3923	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑋 ∈ 𝐴 → 𝑋 ∈ No ))
57		ssel 3923	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑌 ∈ 𝐴 → 𝑌 ∈ No ))
58	56, 57	anim12d 609	. . . . . . . 8 ⊢ (𝐴 ⊆ No → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∈ No ∧ 𝑌 ∈ No )))
59	58	imp 406	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋 ∈ No ∧ 𝑌 ∈ No ))
60		eqtr3 2753	. . . . . . 7 ⊢ ((( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)) → ( bday ‘𝑋) = ( bday ‘𝑌))
61	59, 60	anim12i 613	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴))) → ((𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( bday ‘𝑋) = ( bday ‘𝑌)))
62	61	anasss 466	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)))) → ((𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( bday ‘𝑋) = ( bday ‘𝑌)))
63	62	adantlr 715	. . . 4 ⊢ (((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)))) → ((𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( bday ‘𝑋) = ( bday ‘𝑌)))
64		nodense 27637	. . . . 5 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ (( bday ‘𝑋) = ( bday ‘𝑌) ∧ 𝑋 <s 𝑌)) → ∃𝑤 ∈ No (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌))
65	64	anassrs 467	. . . 4 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ∃𝑤 ∈ No (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌))
66	63, 65	sylan 580	. . 3 ⊢ ((((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)))) ∧ 𝑋 <s 𝑌) → ∃𝑤 ∈ No (( bday ‘𝑤) ∈ ( bday ‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌))
67	55, 66	mtand 815	. 2 ⊢ (((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴)))) → ¬ 𝑋 <s 𝑌)
68	67	ex 412	1 ⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴))) → ¬ 𝑋 <s 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  ∅c0 4282  ∩ cint 4897   class class class wbr 5093  dom cdm 5619  ran crn 5620   “ cima 5622  Oncon0 6312  Fun wfun 6481  ‘cfv 6487   No csur 27584   <s cslt 27585   bday cbday 27586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6315  df-on 6316  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fo 6493  df-fv 6495  df-1o 8391  df-2o 8392  df-no 27587  df-slt 27588  df-bday 27589

This theorem is referenced by:  nocvxmin  27724