Theorem axdense 14027

Description: Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD)

Assertion

Ref	Expression
axdense	|- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> E.x e. No (( bday ` x) e. ( bday ` A) /\ A <s x /\ x <s B))

Distinct variable groups:   x,A   x,B

Proof of Theorem axdense

Step	Hyp	Ref	Expression
1		axdenselem6 14024	. 2 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (A |` |^|{a e. On | (A` a) =/= (B` a)}) e. No )
2		bdayval 13989	. . . . 5 |- ((A |` |^|{a e. On | (A` a) =/= (B` a)}) e. No -> ( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)})) = dom ( A |` |^|{a e. On | (A` a) =/= (B` a)}))
3	1, 2	syl 12	. . . 4 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)})) = dom ( A |` |^|{a e. On | (A` a) =/= (B` a)}))
4		axdenselem5 14023	. . . . . . . 8 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> |^|{a e. On | (A` a) =/= (B` a)} e. ( bday ` A))
5		bdayelon 14017	. . . . . . . . 9 |- ( bday ` A) e. On
6	5	onelssi 3778	. . . . . . . 8 |- (|^|{a e. On | (A` a) =/= (B` a)} e. ( bday ` A) -> |^|{a e. On | (A` a) =/= (B` a)} C_ ( bday ` A))
7	4, 6	syl 12	. . . . . . 7 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> |^|{a e. On | (A` a) =/= (B` a)} C_ ( bday ` A))
8		bdayval 13989	. . . . . . . 8 |- (A e. No -> ( bday ` A) = dom A)
9	8	ad2antrr 440	. . . . . . 7 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ( bday ` A) = dom A)
10	7, 9	sseqtrd 2653	. . . . . 6 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> |^|{a e. On | (A` a) =/= (B` a)} C_ dom A)
11		df-ss 2605	. . . . . 6 |- (|^|{a e. On | (A` a) =/= (B` a)} C_ dom A <-> (|^|{a e. On | (A` a) =/= (B` a)} i^i dom A) = |^|{a e. On | (A` a) =/= (B` a)})
12	10, 11	sylib 215	. . . . 5 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (|^|{a e. On | (A` a) =/= (B` a)} i^i dom A) = |^|{a e. On | (A` a) =/= (B` a)})
13		dmres 4234	. . . . 5 |- dom ( A |` |^|{a e. On | (A` a) =/= (B` a)}) = (|^|{a e. On | (A` a) =/= (B` a)} i^i dom A)
14	12, 13	syl5eq 1940	. . . 4 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> dom ( A |` |^|{a e. On | (A` a) =/= (B` a)}) = |^|{a e. On | (A` a) =/= (B` a)})
15	3, 14	eqtrd 1925	. . 3 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)})) = |^|{a e. On | (A` a) =/= (B` a)})
16	15, 4	eqeltrd 1971	. 2 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)})) e. ( bday ` A))
17		axdenselem4 14022	. . . . 5 |- (((A e. No /\ B e. No ) /\ A <s B) -> |^|{a e. On | (A` a) =/= (B` a)} e. On)
18	17	adantrl 430	. . . 4 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> |^|{a e. On | (A` a) =/= (B` a)} e. On)
19		axdenselem8 14026	. . . . . . . . . . . . 13 |- ((A e. No /\ B e. No /\ ( bday ` A) = ( bday ` B)) -> (A <s B <-> ((A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o)))
20	19	biimpd 170	. . . . . . . . . . . 12 |- ((A e. No /\ B e. No /\ ( bday ` A) = ( bday ` B)) -> (A <s B -> ((A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o)))
21	20	3expia 1069	. . . . . . . . . . 11 |- ((A e. No /\ B e. No ) -> (( bday ` A) = ( bday ` B) -> (A <s B -> ((A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o))))
22	21	imp32 390	. . . . . . . . . 10 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ((A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o))
23	22	simplld 348	. . . . . . . . 9 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o)
24		eqid 1884	. . . . . . . . 9 |- (/) = (/)
25	23, 24	jctir 317	. . . . . . . 8 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ((A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o /\ (/) = (/)))
26	25	3mix1d 13812	. . . . . . 7 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (((A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o /\ (/) = (/)) \/ ((A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o /\ (/) = 2o) \/ ((A` |^|{a e. On | (A` a) =/= (B` a)}) = (/) /\ (/) = 2o)))
27		fvex 4689	. . . . . . . 8 |- (A` |^|{a e. On | (A` a) =/= (B` a)}) e. _V
28		0ex 3446	. . . . . . . 8 |- (/) e. _V
29		2on 5183	. . . . . . . . 9 |- 2o e. On
30	29	elisseti 2301	. . . . . . . 8 |- 2o e. _V
31	27, 28, 28, 30, 30	brtp 13830	. . . . . . 7 |- ((A` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.}(/) <-> (((A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o /\ (/) = (/)) \/ ((A` |^|{a e. On | (A` a) =/= (B` a)}) = 1o /\ (/) = 2o) \/ ((A` |^|{a e. On | (A` a) =/= (B` a)}) = (/) /\ (/) = 2o)))
32	26, 31	sylibr 217	. . . . . 6 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (A` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.}(/))
33	15	fveq2d 4685	. . . . . . 7 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ((A |` |^|{a e. On | (A` a) =/= (B` a)})` ( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)}))) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}))
34		axdenselem2 14020	. . . . . . . 8 |- ((A |` |^|{a e. On | (A` a) =/= (B` a)}) e. No -> ((A |` |^|{a e. On | (A` a) =/= (B` a)})` ( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)}))) = (/))
35	1, 34	syl 12	. . . . . . 7 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ((A |` |^|{a e. On | (A` a) =/= (B` a)})` ( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)}))) = (/))
36	33, 35	eqtr3d 1927	. . . . . 6 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}) = (/))
37	32, 36	breqtrrd 3363	. . . . 5 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (A` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}))
38		fvres 4691	. . . . . . 7 |- (y e. |^|{a e. On | (A` a) =/= (B` a)} -> ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (A` y))
39	38	eqcomd 1889	. . . . . 6 |- (y e. |^|{a e. On | (A` a) =/= (B` a)} -> (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y))
40	39	rgen 2159	. . . . 5 |- A.y e. |^|{a e. On | (A` a) =/= (B` a)} (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y)
41	37, 40	jctil 316	. . . 4 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (A.y e. |^|{a e. On | (A` a) =/= (B` a)} (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) /\ (A` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)})))
42		raleq 2266	. . . . . 6 |- (x = |^|{a e. On | (A` a) =/= (B` a)} -> (A.y e. x (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) <-> A.y e. |^|{a e. On | (A` a) =/= (B` a)} (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y)))
43		fveq2 4681	. . . . . . 7 |- (x = |^|{a e. On | (A` a) =/= (B` a)} -> (A` x) = (A` |^|{a e. On | (A` a) =/= (B` a)}))
44		fveq2 4681	. . . . . . 7 |- (x = |^|{a e. On | (A` a) =/= (B` a)} -> ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}))
45	43, 44	breq12d 3351	. . . . . 6 |- (x = |^|{a e. On | (A` a) =/= (B` a)} -> ((A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x) <-> (A` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)})))
46	42, 45	anbi12d 690	. . . . 5 |- (x = |^|{a e. On | (A` a) =/= (B` a)} -> ((A.y e. x (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x)) <-> (A.y e. |^|{a e. On | (A` a) =/= (B` a)} (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) /\ (A` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}))))
47	46	rcla4ev 2381	. . . 4 |- ((|^|{a e. On | (A` a) =/= (B` a)} e. On /\ (A.y e. |^|{a e. On | (A` a) =/= (B` a)} (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) /\ (A` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}))) -> E.x e. On (A.y e. x (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x)))
48	18, 41, 47	syl11anc 524	. . 3 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> E.x e. On (A.y e. x (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x)))
49		simpll 448	. . . 4 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> A e. No )
50		sltval 13988	. . . 4 |- ((A e. No /\ (A |` |^|{a e. On | (A` a) =/= (B` a)}) e. No ) -> (A <s (A |` |^|{a e. On | (A` a) =/= (B` a)}) <-> E.x e. On (A.y e. x (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x))))
51	49, 1, 50	syl11anc 524	. . 3 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (A <s (A |` |^|{a e. On | (A` a) =/= (B` a)}) <-> E.x e. On (A.y e. x (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x))))
52	48, 51	mpbird 213	. 2 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> A <s (A |` |^|{a e. On | (A` a) =/= (B` a)}))
53	39	adantl 424	. . . . . 6 |- ((((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) /\ y e. |^|{a e. On | (A` a) =/= (B` a)}) -> (A` y) = ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y))
54		axdenselem7 14025	. . . . . . 7 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (y e. |^|{a e. On | (A` a) =/= (B` a)} -> (A` y) = (B` y)))
55	54	imp 377	. . . . . 6 |- ((((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) /\ y e. |^|{a e. On | (A` a) =/= (B` a)}) -> (A` y) = (B` y))
56	53, 55	eqtr3d 1927	. . . . 5 |- ((((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) /\ y e. |^|{a e. On | (A` a) =/= (B` a)}) -> ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y))
57	56	r19.21aiva 2176	. . . 4 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> A.y e. |^|{a e. On | (A` a) =/= (B` a)} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y))
58	22	simprd 352	. . . . . . . 8 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o)
59	58, 24	jctil 316	. . . . . . 7 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ((/) = (/) /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o))
60	59	3mix3d 13814	. . . . . 6 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (((/) = 1o /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = (/)) \/ ((/) = 1o /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o) \/ ((/) = (/) /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o)))
61		fvex 4689	. . . . . . 7 |- (B` |^|{a e. On | (A` a) =/= (B` a)}) e. _V
62	28, 61, 28, 30, 30	brtp 13830	. . . . . 6 |- ((/){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` |^|{a e. On | (A` a) =/= (B` a)}) <-> (((/) = 1o /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = (/)) \/ ((/) = 1o /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o) \/ ((/) = (/) /\ (B` |^|{a e. On | (A` a) =/= (B` a)}) = 2o)))
63	60, 62	sylibr 217	. . . . 5 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (/){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` |^|{a e. On | (A` a) =/= (B` a)}))
64	36, 63	eqbrtrd 3357	. . . 4 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` |^|{a e. On | (A` a) =/= (B` a)}))
65		raleq 2266	. . . . . 6 |- (x = |^|{a e. On | (A` a) =/= (B` a)} -> (A.y e. x ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y) <-> A.y e. |^|{a e. On | (A` a) =/= (B` a)} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y)))
66		fveq2 4681	. . . . . . 7 |- (x = |^|{a e. On | (A` a) =/= (B` a)} -> (B` x) = (B` |^|{a e. On | (A` a) =/= (B` a)}))
67	44, 66	breq12d 3351	. . . . . 6 |- (x = |^|{a e. On | (A` a) =/= (B` a)} -> (((A |` |^|{a e. On | (A` a) =/= (B` a)})` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x) <-> ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` |^|{a e. On | (A` a) =/= (B` a)})))
68	65, 67	anbi12d 690	. . . . 5 |- (x = |^|{a e. On | (A` a) =/= (B` a)} -> ((A.y e. x ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y) /\ ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x)) <-> (A.y e. |^|{a e. On | (A` a) =/= (B` a)} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y) /\ ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` |^|{a e. On | (A` a) =/= (B` a)}))))
69	68	rcla4ev 2381	. . . 4 |- ((|^|{a e. On | (A` a) =/= (B` a)} e. On /\ (A.y e. |^|{a e. On | (A` a) =/= (B` a)} ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y) /\ ((A |` |^|{a e. On | (A` a) =/= (B` a)})` |^|{a e. On | (A` a) =/= (B` a)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` |^|{a e. On | (A` a) =/= (B` a)}))) -> E.x e. On (A.y e. x ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y) /\ ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x)))
70	18, 57, 64, 69	syl12anc 1098	. . 3 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> E.x e. On (A.y e. x ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y) /\ ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x)))
71		simplr 449	. . . 4 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> B e. No )
72		sltval 13988	. . . 4 |- (((A |` |^|{a e. On | (A` a) =/= (B` a)}) e. No /\ B e. No ) -> ((A |` |^|{a e. On | (A` a) =/= (B` a)}) <s B <-> E.x e. On (A.y e. x ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y) /\ ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x))))
73	1, 71, 72	syl11anc 524	. . 3 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> ((A |` |^|{a e. On | (A` a) =/= (B` a)}) <s B <-> E.x e. On (A.y e. x ((A |` |^|{a e. On | (A` a) =/= (B` a)})` y) = (B` y) /\ ((A |` |^|{a e. On | (A` a) =/= (B` a)})` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x))))
74	70, 73	mpbird 213	. 2 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> (A |` |^|{a e. On | (A` a) =/= (B` a)}) <s B)
75		fveq2 4681	. . . . 5 |- (x = (A |` |^|{a e. On | (A` a) =/= (B` a)}) -> ( bday ` x) = ( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)})))
76	75	eleq1d 1963	. . . 4 |- (x = (A |` |^|{a e. On | (A` a) =/= (B` a)}) -> (( bday ` x) e. ( bday ` A) <-> ( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)})) e. ( bday ` A)))
77		breq2 3342	. . . 4 |- (x = (A |` |^|{a e. On | (A` a) =/= (B` a)}) -> (A <s x <-> A <s (A |` |^|{a e. On | (A` a) =/= (B` a)})))
78		breq1 3341	. . . 4 |- (x = (A |` |^|{a e. On | (A` a) =/= (B` a)}) -> (x <s B <-> (A |` |^|{a e. On | (A` a) =/= (B` a)}) <s B))
79	76, 77, 78	3anbi123d 1168	. . 3 |- (x = (A |` |^|{a e. On | (A` a) =/= (B` a)}) -> ((( bday ` x) e. ( bday ` A) /\ A <s x /\ x <s B) <-> (( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)})) e. ( bday ` A) /\ A <s (A |` |^|{a e. On | (A` a) =/= (B` a)}) /\ (A |` |^|{a e. On | (A` a) =/= (B` a)}) <s B)))
80	79	rcla4ev 2381	. 2 |- (((A |` |^|{a e. On | (A` a) =/= (B` a)}) e. No /\ (( bday ` (A |` |^|{a e. On | (A` a) =/= (B` a)})) e. ( bday ` A) /\ A <s (A |` |^|{a e. On | (A` a) =/= (B` a)}) /\ (A |` |^|{a e. On | (A` a) =/= (B` a)}) <s B)) -> E.x e. No (( bday ` x) e. ( bday ` A) /\ A <s x /\ x <s B))
81	1, 16, 52, 74, 80	syl13anc 1102	1 |- (((A e. No /\ B e. No ) /\ (( bday ` A) = ( bday ` B) /\ A <s B)) -> E.x e. No (( bday ` x) e. ( bday ` A) /\ A <s x /\ x <s B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   \/ w3o 857   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   i^i cin 2592   C_ wss 2593  (/)c0 2875  <.cop 3046  {ctp 3051  |^|cint 3214   class class class wbr 3338  Oncon0 3657  dom cdm 3986   |` cres 3988  ` cfv 3998  1oc1o 5172  2oc2o 5173   No csur 13981   <s cslt 13982   bday cbday 13983

This theorem is referenced by:  nocvxminlem 14028

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-mpt 5006  df-1o 5177  df-2o 5178  df-no 13984  df-slt 13985  df-bday 13986

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