Theorem nocvxminlem 33972

Description: Lemma for nocvxmin 33973. 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 5077	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → (𝑥 <s 𝑧 ↔ 𝑋 <s 𝑧))
2	1	anbi1d 630	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦)))
3	2	imbi1d 342	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) ↔ ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)))
4	3	ralbidv 3112	. . . . . . . . . . 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 5078	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑧 <s 𝑦 ↔ 𝑧 <s 𝑌))
6	5	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌)))
7	6	imbi1d 342	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → (((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) ↔ ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴)))
8	7	ralbidv 3112	. . . . . . . . . . 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 3570	. . . . . . . . . 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 5078	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑤))
11		breq1 5077	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑧 <s 𝑌 ↔ 𝑤 <s 𝑌))
12	10, 11	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) ↔ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
13		eleq1w 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
14	12, 13	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) ↔ ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → 𝑤 ∈ 𝐴)))
15	14	rspcv 3557	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ No → (∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → 𝑤 ∈ 𝐴)))
16		bdaydm 33969	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom bday = No
17	16	sseq2i 3950	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
18		bdayfun 33967	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Fun bday
19		funfvima2 7107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday ) → (𝑤 ∈ 𝐴 → ( bday ‘𝑤) ∈ ( bday “ 𝐴)))
20	18, 19	mpan 687	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ dom bday → (𝑤 ∈ 𝐴 → ( bday ‘𝑤) ∈ ( bday “ 𝐴)))
21	17, 20	sylbir 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → ( bday ‘𝑤) ∈ ( bday “ 𝐴)))
22	21	imp 407	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ( bday ‘𝑤) ∈ ( bday “ 𝐴))
23		intss1 4894	. . . . . . . . . . . . . . . . . . 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 5980	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday “ 𝐴) ⊆ ran bday
26		bdayrn 33970	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran bday = On
27	25, 26	sseqtri 3957	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( bday “ 𝐴) ⊆ On
28	22	ne0d 4269	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ( bday “ 𝐴) ≠ ∅)
29		oninton 7645	. . . . . . . . . . . . . . . . . . . 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 33971	. . . . . . . . . . . . . . . . . . 19 ⊢ ( bday ‘𝑤) ∈ On
32		ontri1 6300	. . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ¬ ( bday ‘𝑤) ∈ ∩ ( bday “ 𝐴))
35	34	ex 413	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → ¬ ( bday ‘𝑤) ∈ ∩ ( bday “ 𝐴)))
36		eleq2 2827	. . . . . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 408	. . . . . . 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 427	. . . . . 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 400	. . . . . 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 277	. . . . 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 233	. . . 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 3198	. . 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 3914	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑋 ∈ 𝐴 → 𝑋 ∈ No ))
57		ssel 3914	. . . . . . . . 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 407	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋 ∈ No ∧ 𝑌 ∈ No ))
60		eqtr3 2764	. . . . . . 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 467	. . . . 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 712	. . . 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 33895	. . . . 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 468	. . . 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 813	. 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 413	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∅c0 4256  ∩ cint 4879   class class class wbr 5074  dom cdm 5589  ran crn 5590   “ cima 5592  Oncon0 6266  Fun wfun 6427  ‘cfv 6433   No csur 33843   <s cslt 33844   bday cbday 33845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848

This theorem is referenced by:  nocvxmin  33973