Theorem nocvxminlem 28173

Description: Lemma for nocvxmin 28174. 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 5109	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → (𝑥 <s 𝑧 ↔ 𝑋 <s 𝑧))
2	1	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦)))
3	2	imbi1d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) ↔ ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)))
4	3	ralbidv 3189	. . . . . . . . . . 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 5110	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑧 <s 𝑦 ↔ 𝑧 <s 𝑌))
6	5	anbi2d 642	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌)))
7	6	imbi1d 344	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → (((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴) ↔ ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴)))
8	7	ralbidv 3189	. . . . . . . . . . 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 3590	. . . . . . . . . 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 5110	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑤))
11		breq1 5109	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑧 <s 𝑌 ↔ 𝑤 <s 𝑌))
12	10, 11	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) ↔ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
13		eleq1w 2847	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
14	12, 13	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) ↔ ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → 𝑤 ∈ 𝐴)))
15	14	rspcv 3576	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ No → (∀𝑧 ∈ No ((𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌) → 𝑧 ∈ 𝐴) → ((𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) → 𝑤 ∈ 𝐴)))
16		bdaydm 28168	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom bday = No
17	16	sseq2i 3942	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No)
18		bdayfun 28166	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Fun bday
19		funfvima2 7263	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday) → (𝑤 ∈ 𝐴 → (bday‘𝑤) ∈ (bday “ 𝐴)))
20	18, 19	mpan 703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ dom bday → (𝑤 ∈ 𝐴 → (bday‘𝑤) ∈ (bday “ 𝐴)))
21	17, 20	sylbir 238	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → (bday‘𝑤) ∈ (bday “ 𝐴)))
22	21	imp 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → (bday‘𝑤) ∈ (bday “ 𝐴))
23		intss1 4926	. . . . . . . . . . . . . . . . . . 19 ⊢ ((bday‘𝑤) ∈ (bday “ 𝐴) → ∩ (bday “ 𝐴) ⊆ (bday‘𝑤))
24	22, 23	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ∩ (bday “ 𝐴) ⊆ (bday‘𝑤))
25		imassrn 6211	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (bday “ 𝐴) ⊆ ran bday
26		bdayrn 28170	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran bday = On
27	25, 26	sseqtri 3961	. . . . . . . . . . . . . . . . . . . 20 ⊢ (bday “ 𝐴) ⊆ On
28	22	ne0d 4291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → (bday “ 𝐴) ≠ ∅)
29		oninton 7836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((bday “ 𝐴) ⊆ On ∧ (bday “ 𝐴) ≠ ∅) → ∩ (bday “ 𝐴) ∈ On)
30	27, 28, 29	sylancr 599	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ∩ (bday “ 𝐴) ∈ On)
31		bdayon 28171	. . . . . . . . . . . . . . . . . . 19 ⊢ (bday‘𝑤) ∈ On
32		ontri1 6418	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∩ (bday “ 𝐴) ∈ On ∧ (bday‘𝑤) ∈ On) → (∩ (bday “ 𝐴) ⊆ (bday‘𝑤) ↔ ¬ (bday‘𝑤) ∈ ∩ (bday “ 𝐴)))
33	30, 31, 32	sylancl 598	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → (∩ (bday “ 𝐴) ⊆ (bday‘𝑤) ↔ ¬ (bday‘𝑤) ∈ ∩ (bday “ 𝐴)))
34	24, 33	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴) → ¬ (bday‘𝑤) ∈ ∩ (bday “ 𝐴))
35	34	ex 418	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → ¬ (bday‘𝑤) ∈ ∩ (bday “ 𝐴)))
36		eleq2 2853	. . . . . . . . . . . . . . . . . 18 ⊢ ((bday‘𝑋) = ∩ (bday “ 𝐴) → ((bday‘𝑤) ∈ (bday‘𝑋) ↔ (bday‘𝑤) ∈ ∩ (bday “ 𝐴)))
37	36	notbid 321	. . . . . . . . . . . . . . . . 17 ⊢ ((bday‘𝑋) = ∩ (bday “ 𝐴) → (¬ (bday‘𝑤) ∈ (bday‘𝑋) ↔ ¬ (bday‘𝑤) ∈ ∩ (bday “ 𝐴)))
38	37	biimprcd 253	. . . . . . . . . . . . . . . 16 ⊢ (¬ (bday‘𝑤) ∈ ∩ (bday “ 𝐴) → ((bday‘𝑋) = ∩ (bday “ 𝐴) → ¬ (bday‘𝑤) ∈ (bday‘𝑋)))
39	35, 38	syl6 36	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → ((bday‘𝑋) = ∩ (bday “ 𝐴) → ¬ (bday‘𝑤) ∈ (bday‘𝑋))))
40	39	com3l 90	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝐴 → ((bday‘𝑋) = ∩ (bday “ 𝐴) → (𝐴 ⊆ No → ¬ (bday‘𝑤) ∈ (bday‘𝑋))))
41	40	adantrd 497	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝐴 → (((bday‘𝑋) = ∩ (bday “ 𝐴) ∧ (bday‘𝑌) = ∩ (bday “ 𝐴)) → (𝐴 ⊆ No → ¬ (bday‘𝑤) ∈ (bday‘𝑋))))
42	15, 41	syl8 77	. . . . . . . . . . . 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 99	. . . . . . . . . . 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 93	. . . . . . . . . 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 36	. . . . . . . . 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 90	. . . . . . . 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 413	. . . . . . 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 432	. . . . . 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 135	. . . . 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 1111	. . . . . . 7 ⊢ (((bday‘𝑤) ∈ (bday‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) ↔ ((bday‘𝑤) ∈ (bday‘𝑋) ∧ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
51	50	notbii 323	. . . . . 6 ⊢ (¬ ((bday‘𝑤) ∈ (bday‘𝑋) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌) ↔ ¬ ((bday‘𝑤) ∈ (bday‘𝑋) ∧ (𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌)))
52		imnan 405	. . . . . 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 281	. . . . 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 237	. . . 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 3161	. . 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 3907	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑋 ∈ 𝐴 → 𝑋 ∈ No))
57		ssel 3907	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑌 ∈ 𝐴 → 𝑌 ∈ No))
58	56, 57	anim12d 621	. . . . . . . 8 ⊢ (𝐴 ⊆ No → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∈ No ∧ 𝑌 ∈ No)))
59	58	imp 412	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋 ∈ No ∧ 𝑌 ∈ No))
60		eqtr3 2785	. . . . . . 7 ⊢ (((bday‘𝑋) = ∩ (bday “ 𝐴) ∧ (bday‘𝑌) = ∩ (bday “ 𝐴)) → (bday‘𝑋) = (bday‘𝑌))
61	59, 60	anim12i 625	. . . . . 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 472	. . . . 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 728	. . . 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 28082	. . . . 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 473	. . . 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 592	. . 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 828	. 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 418	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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281  ∩ cint 4910   class class class wbr 5106  dom cdm 5656  ran crn 5657   “ cima 5659  Oncon0 6383  Fun wfun 6555  ‘cfv 6561  Nocsur 28030   <s clts 28031  bdaycbday 28032

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035

This theorem is used by:  nocvxmin  28174