Theorem scutbdaylt 31922

Description: If a surreal lies in a gap and is not equal to the cut, its birthday is greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.)

Assertion

Ref	Expression
scutbdaylt	|- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> ( bday ` ( A |s B ) ) e. ( bday ` X ) )

Proof of Theorem scutbdaylt

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2l 1087	. . . . 5 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> A < <s { X } )
2		simp2r 1088	. . . . 5 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> { X } < <s B )
3		snnzg 4308	. . . . . 6 |- ( X e. No -> { X } =/= (/) )
4	3	3ad2ant1 1082	. . . . 5 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> { X } =/= (/) )
5		sslttr 31914	. . . . 5 |- ( ( A < <s { X } /\ { X } < <s B /\ { X } =/= (/) ) -> A < <s B )
6	1, 2, 4, 5	syl3anc 1326	. . . 4 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> A < <s B )
7		scutbday 31913	. . . 4 |- ( A < <s B -> ( bday ` ( A |s B ) ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
8	6, 7	syl 17	. . 3 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> ( bday ` ( A |s B ) ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
9		bdayfn 31889	. . . . 5 |- bday Fn No
10		ssrab2 3687	. . . . 5 |- { y e. No | ( A < <s { y } /\ { y } < <s B ) } C_ No
11		simp1 1061	. . . . . 6 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> X e. No )
12		simp2 1062	. . . . . 6 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> ( A < <s { X } /\ { X } < <s B ) )
13		sneq 4187	. . . . . . . . 9 |- ( y = X -> { y } = { X } )
14	13	breq2d 4665	. . . . . . . 8 |- ( y = X -> ( A < <s { y } <-> A < <s { X } ) )
15	13	breq1d 4663	. . . . . . . 8 |- ( y = X -> ( { y } < <s B <-> { X } < <s B ) )
16	14, 15	anbi12d 747	. . . . . . 7 |- ( y = X -> ( ( A < <s { y } /\ { y } < <s B ) <-> ( A < <s { X } /\ { X } < <s B ) ) )
17	16	elrab 3363	. . . . . 6 |- ( X e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } <-> ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) )
18	11, 12, 17	sylanbrc 698	. . . . 5 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> X e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } )
19		fnfvima 6496	. . . . 5 |- ( ( bday Fn No /\ { y e. No | ( A < <s { y } /\ { y } < <s B ) } C_ No /\ X e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) -> ( bday ` X ) e. ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
20	9, 10, 18, 19	mp3an12i 1428	. . . 4 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> ( bday ` X ) e. ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
21		intss1 4492	. . . 4 |- ( ( bday ` X ) e. ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) -> |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) C_ ( bday ` X ) )
22	20, 21	syl 17	. . 3 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) C_ ( bday ` X ) )
23	8, 22	eqsstrd 3639	. 2 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> ( bday ` ( A |s B ) ) C_ ( bday ` X ) )
24		simprl 794	. . . . . . . . . . . 12 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> A < <s { X } )
25		simprr 796	. . . . . . . . . . . 12 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> { X } < <s B )
26	3	adantr 481	. . . . . . . . . . . 12 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> { X } =/= (/) )
27	24, 25, 26, 5	syl3anc 1326	. . . . . . . . . . 11 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> A < <s B )
28	27, 7	syl 17	. . . . . . . . . 10 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> ( bday ` ( A |s B ) ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
29	28	eqeq1d 2624	. . . . . . . . 9 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> ( ( bday ` ( A |s B ) ) = ( bday ` X ) <-> |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) = ( bday ` X ) ) )
30		eqcom 2629	. . . . . . . . 9 |- ( |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) = ( bday ` X ) <-> ( bday ` X ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
31	29, 30	syl6bb 276	. . . . . . . 8 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> ( ( bday ` ( A |s B ) ) = ( bday ` X ) <-> ( bday ` X ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
32	31	biimpa 501	. . . . . . 7 |- ( ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) /\ ( bday ` ( A |s B ) ) = ( bday ` X ) ) -> ( bday ` X ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
33	17	biimpri 218	. . . . . . . 8 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> X e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } )
34	27	adantr 481	. . . . . . . . 9 |- ( ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) /\ ( bday ` ( A |s B ) ) = ( bday ` X ) ) -> A < <s B )
35		conway 31910	. . . . . . . . 9 |- ( A < <s B -> E! x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
36	34, 35	syl 17	. . . . . . . 8 |- ( ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) /\ ( bday ` ( A |s B ) ) = ( bday ` X ) ) -> E! x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
37		fveq2 6191	. . . . . . . . . . 11 |- ( x = X -> ( bday ` x ) = ( bday ` X ) )
38	37	eqeq1d 2624	. . . . . . . . . 10 |- ( x = X -> ( ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) <-> ( bday ` X ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
39	38	riota2 6633	. . . . . . . . 9 |- ( ( X e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } /\ E! x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) -> ( ( bday ` X ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) <-> ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) = X ) )
40		eqcom 2629	. . . . . . . . 9 |- ( ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) = X <-> X = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
41	39, 40	syl6bb 276	. . . . . . . 8 |- ( ( X e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } /\ E! x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) -> ( ( bday ` X ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) <-> X = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) ) )
42	33, 36, 41	syl2an2r 876	. . . . . . 7 |- ( ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) /\ ( bday ` ( A |s B ) ) = ( bday ` X ) ) -> ( ( bday ` X ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) <-> X = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) ) )
43	32, 42	mpbid 222	. . . . . 6 |- ( ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) /\ ( bday ` ( A |s B ) ) = ( bday ` X ) ) -> X = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
44		scutval 31911	. . . . . . 7 |- ( A < <s B -> ( A |s B ) = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
45	34, 44	syl 17	. . . . . 6 |- ( ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) /\ ( bday ` ( A |s B ) ) = ( bday ` X ) ) -> ( A |s B ) = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
46	43, 45	eqtr4d 2659	. . . . 5 |- ( ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) /\ ( bday ` ( A |s B ) ) = ( bday ` X ) ) -> X = ( A |s B ) )
47	46	ex 450	. . . 4 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> ( ( bday ` ( A |s B ) ) = ( bday ` X ) -> X = ( A |s B ) ) )
48	47	necon3d 2815	. . 3 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) ) -> ( X =/= ( A |s B ) -> ( bday ` ( A |s B ) ) =/= ( bday ` X ) ) )
49	48	3impia 1261	. 2 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> ( bday ` ( A |s B ) ) =/= ( bday ` X ) )
50		bdayelon 31892	. . 3 |- ( bday ` ( A |s B ) ) e. On
51		bdayelon 31892	. . 3 |- ( bday ` X ) e. On
52		onelpss 5764	. . 3 |- ( ( ( bday ` ( A |s B ) ) e. On /\ ( bday ` X ) e. On ) -> ( ( bday ` ( A |s B ) ) e. ( bday ` X ) <-> ( ( bday ` ( A |s B ) ) C_ ( bday ` X ) /\ ( bday ` ( A |s B ) ) =/= ( bday ` X ) ) ) )
53	50, 51, 52	mp2an 708	. 2 |- ( ( bday ` ( A |s B ) ) e. ( bday ` X ) <-> ( ( bday ` ( A |s B ) ) C_ ( bday ` X ) /\ ( bday ` ( A |s B ) ) =/= ( bday ` X ) ) )
54	23, 49, 53	sylanbrc 698	1 |- ( ( X e. No /\ ( A < <s { X } /\ { X } < <s B ) /\ X =/= ( A |s B ) ) -> ( bday ` ( A |s B ) ) e. ( bday ` X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E!wreu 2914   {crab 2916   C_ wss 3574   (/)c0 3915   {csn 4177   |^|cint 4475   class class class wbr 4653   "cima 5117   Oncon0 5723   Fn wfn 5883   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Nocsur 31793   bdaycbday 31795   < <scsslt 31896   |scscut 31898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798  df-sslt 31897  df-scut 31899

This theorem is referenced by:  slerec  31923