Theorem scutbdaybnd 31921

Description: An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.)

Assertion

Ref	Expression
scutbdaybnd	|- ( A < <s B -> ( bday ` ( A |s B ) ) C_ suc U. ( bday " ( A u. B ) ) )

Proof of Theorem scutbdaybnd

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

Step	Hyp	Ref	Expression
1		etasslt 31920	. 2 |- ( A < <s B -> E. x e. No ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) )
2		scutbday 31913	. . . . . 6 |- ( A < <s B -> ( bday ` ( A |s B ) ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
3	2	adantr 481	. . . . 5 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> ( bday ` ( A |s B ) ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
4		bdayfn 31889	. . . . . . 7 |- bday Fn No
5		ssrab2 3687	. . . . . . 7 |- { y e. No | ( A < <s { y } /\ { y } < <s B ) } C_ No
6		simprl 794	. . . . . . . 8 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> x e. No )
7		simprr1 1109	. . . . . . . . 9 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> A < <s { x } )
8		simprr2 1110	. . . . . . . . 9 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> { x } < <s B )
9	7, 8	jca 554	. . . . . . . 8 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> ( A < <s { x } /\ { x } < <s B ) )
10		sneq 4187	. . . . . . . . . . 11 |- ( y = x -> { y } = { x } )
11	10	breq2d 4665	. . . . . . . . . 10 |- ( y = x -> ( A < <s { y } <-> A < <s { x } ) )
12	10	breq1d 4663	. . . . . . . . . 10 |- ( y = x -> ( { y } < <s B <-> { x } < <s B ) )
13	11, 12	anbi12d 747	. . . . . . . . 9 |- ( y = x -> ( ( A < <s { y } /\ { y } < <s B ) <-> ( A < <s { x } /\ { x } < <s B ) ) )
14	13	elrab 3363	. . . . . . . 8 |- ( x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } <-> ( x e. No /\ ( A < <s { x } /\ { x } < <s B ) ) )
15	6, 9, 14	sylanbrc 698	. . . . . . 7 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } )
16		fnfvima 6496	. . . . . . 7 |- ( ( 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 ) } ) )
17	4, 5, 15, 16	mp3an12i 1428	. . . . . 6 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> ( bday ` x ) e. ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
18		intss1 4492	. . . . . 6 |- ( ( 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 ) )
19	17, 18	syl 17	. . . . 5 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) C_ ( bday ` x ) )
20	3, 19	eqsstrd 3639	. . . 4 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> ( bday ` ( A |s B ) ) C_ ( bday ` x ) )
21		simprr3 1111	. . . 4 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) )
22	20, 21	sstrd 3613	. . 3 |- ( ( A < <s B /\ ( x e. No /\ ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) ) ) -> ( bday ` ( A |s B ) ) C_ suc U. ( bday " ( A u. B ) ) )
23	22	rexlimdvaa 3032	. 2 |- ( A < <s B -> ( E. x e. No ( A < <s { x } /\ { x } < <s B /\ ( bday ` x ) C_ suc U. ( bday " ( A u. B ) ) ) -> ( bday ` ( A |s B ) ) C_ suc U. ( bday " ( A u. B ) ) ) )
24	1, 23	mpd 15	1 |- ( A < <s B -> ( bday ` ( A |s B ) ) C_ suc U. ( bday " ( A u. B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   u. cun 3572   C_ wss 3574   {csn 4177   U.cuni 4436   |^|cint 4475   class class class wbr 4653   "cima 5117   suc csuc 5725   Fn wfn 5883   ` cfv 5888  (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: (None)