Theorem nocvxminlem 27689

Description: Lemma for nocvxmin 27690. 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 5110	. . . . . . . . . . . . . 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 3156	. . . . . . . . . . 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 5111	. . . . . . . . . . . . . 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 3156	. . . . . . . . . . 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 3599	. . . . . . . . . 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 5111	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑤))
11		breq1 5110	. . . . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ No → (∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → 𝑤 ∈ 𝐴)))
16		bdaydm 27686	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom bday = No
17	16	sseq2i 3976	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
18		bdayfun 27684	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Fun bday
19		funfvima2 7205	. . . . . . . . . . . . . . . . . . . . . 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 4927	. . . . . . . . . . . . . . . . . . 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 6042	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday “ 𝐴) ⊆ ran bday
26		bdayrn 27687	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran bday = On
27	25, 26	sseqtri 3995	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( bday “ 𝐴) ⊆ On
28	22	ne0d 4305	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ( bday “ 𝐴) ≠ ∅)
29		oninton 7771	. . . . . . . . . . . . . . . . . . . 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 27688	. . . . . . . . . . . . . . . . . . 19 ⊢ ( bday ‘𝑤) ∈ On
32		ontri1 6366	. . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . 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 3128	. . 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 3940	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑋 ∈ 𝐴 → 𝑋 ∈ No ))
57		ssel 3940	. . . . . . . . 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 2751	. . . . . . 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 27604	. . . . 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 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  ∅c0 4296  ∩ cint 4910   class class class wbr 5107  dom cdm 5638  ran crn 5639   “ cima 5641  Oncon0 6332  Fun wfun 6505  ‘cfv 6511   No csur 27551   <s cslt 27552   bday cbday 27553

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-1o 8434  df-2o 8435  df-no 27554  df-slt 27555  df-bday 27556

This theorem is referenced by:  nocvxmin  27690