Theorem nobndlem8 30659

Description: Lemma for nobndup 30660 and nobnddown 30661. Bound the birthday of ( C X. { S } ) above. (Contributed by Scott Fenton, 10-Apr-2017.)

Hypotheses

Ref	Expression
nobndlem8.1	|- S e. { 1o , 2o }
nobndlem8.2	|- C = |^| { a e. On | A. n e. F E. b e. a ( n ` b ) =/= S }

Assertion

Ref	Expression
nobndlem8	|- ( ( F C_ No /\ F e. A ) -> ( bday ` ( C X. { S } ) ) C_ suc U. ( bday " F ) )

Distinct variable groups:   F, a, b, n   S, a, b

Allowed substitution hints:   A( n, a, b)   C( n, a, b)   S( n)

Proof of Theorem nobndlem8

Step	Hyp	Ref	Expression
1		elex 3040	. 2 |- ( F e. A -> F e. _V )
2		nobndlem8.1	. . . . 5 |- S e. { 1o , 2o }
3		nobndlem8.2	. . . . 5 |- C = |^| { a e. On | A. n e. F E. b e. a ( n ` b ) =/= S }
4	2, 3	nobndlem3 30654	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) = C )
5	4, 3	syl6eq 2521	. . 3 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) = |^| { a e. On | A. n e. F E. b e. a ( n ` b ) =/= S } )
6		nobndlem1 30652	. . . . 5 |- ( F e. _V -> suc U. ( bday " F ) e. On )
7	6	adantl 473	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> suc U. ( bday " F ) e. On )
8		bdayfn 30639	. . . . . . . . . . 11 |- bday Fn No
9		fnfvima 6161	. . . . . . . . . . 11 |- ( ( bday Fn No /\ F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
10	8, 9	mp3an1 1377	. . . . . . . . . 10 |- ( ( F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
11	10	3adant2 1049	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
12		elssuni 4219	. . . . . . . . 9 |- ( ( bday ` n ) e. ( bday " F ) -> ( bday ` n ) C_ U. ( bday " F ) )
13	11, 12	syl 17	. . . . . . . 8 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) C_ U. ( bday " F ) )
14		bdayelon 30640	. . . . . . . . 9 |- ( bday ` n ) e. On
15		sucelon 6663	. . . . . . . . . . 11 |- ( U. ( bday " F ) e. On <-> suc U. ( bday " F ) e. On )
16	6, 15	sylibr 217	. . . . . . . . . 10 |- ( F e. _V -> U. ( bday " F ) e. On )
17	16	3ad2ant2 1052	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> U. ( bday " F ) e. On )
18		onsssuc 5517	. . . . . . . . 9 |- ( ( ( bday ` n ) e. On /\ U. ( bday " F ) e. On ) -> ( ( bday ` n ) C_ U. ( bday " F ) <-> ( bday ` n ) e. suc U. ( bday " F ) ) )
19	14, 17, 18	sylancr 676	. . . . . . . 8 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( ( bday ` n ) C_ U. ( bday " F ) <-> ( bday ` n ) e. suc U. ( bday " F ) ) )
20	13, 19	mpbid 215	. . . . . . 7 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) e. suc U. ( bday " F ) )
21		ssel2 3413	. . . . . . . . 9 |- ( ( F C_ No /\ n e. F ) -> n e. No )
22		fvnobday 30642	. . . . . . . . . 10 |- ( n e. No -> ( n ` ( bday ` n ) ) = (/) )
23	2	nosgnn0i 30617	. . . . . . . . . . 11 |- (/) =/= S
24	23	a1i 11	. . . . . . . . . 10 |- ( n e. No -> (/) =/= S )
25	22, 24	eqnetrd 2710	. . . . . . . . 9 |- ( n e. No -> ( n ` ( bday ` n ) ) =/= S )
26	21, 25	syl 17	. . . . . . . 8 |- ( ( F C_ No /\ n e. F ) -> ( n ` ( bday ` n ) ) =/= S )
27	26	3adant2 1049	. . . . . . 7 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( n ` ( bday ` n ) ) =/= S )
28		fveq2 5879	. . . . . . . . 9 |- ( b = ( bday ` n ) -> ( n ` b ) = ( n ` ( bday ` n ) ) )
29	28	neeq1d 2702	. . . . . . . 8 |- ( b = ( bday ` n ) -> ( ( n ` b ) =/= S <-> ( n ` ( bday ` n ) ) =/= S ) )
30	29	rspcev 3136	. . . . . . 7 |- ( ( ( bday ` n ) e. suc U. ( bday " F ) /\ ( n ` ( bday ` n ) ) =/= S ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
31	20, 27, 30	syl2anc 673	. . . . . 6 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
32	31	3expa 1231	. . . . 5 |- ( ( ( F C_ No /\ F e. _V ) /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
33	32	ralrimiva 2809	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> A. n e. F E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
34		rexeq 2974	. . . . . 6 |- ( a = suc U. ( bday " F ) -> ( E. b e. a ( n ` b ) =/= S <-> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S ) )
35	34	ralbidv 2829	. . . . 5 |- ( a = suc U. ( bday " F ) -> ( A. n e. F E. b e. a ( n ` b ) =/= S <-> A. n e. F E. b e. suc U. ( bday " F ) ( n ` b ) =/= S ) )
36	35	intminss 4252	. . . 4 |- ( ( suc U. ( bday " F ) e. On /\ A. n e. F E. b e. suc U. ( bday " F ) ( n ` b ) =/= S ) -> |^| { a e. On | A. n e. F E. b e. a ( n ` b ) =/= S } C_ suc U. ( bday " F ) )
37	7, 33, 36	syl2anc 673	. . 3 |- ( ( F C_ No /\ F e. _V ) -> |^| { a e. On | A. n e. F E. b e. a ( n ` b ) =/= S } C_ suc U. ( bday " F ) )
38	5, 37	eqsstrd 3452	. 2 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) C_ suc U. ( bday " F ) )
39	1, 38	sylan2 482	1 |- ( ( F C_ No /\ F e. A ) -> ( bday ` ( C X. { S } ) ) C_ suc U. ( bday " F ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   C_ wss 3390   (/)c0 3722   {csn 3959   {cpr 3961   U.cuni 4190   |^|cint 4226   X. cxp 4837   "cima 4842   Oncon0 5430   suc csuc 5432   Fn wfn 5584   ` cfv 5589   1oc1o 7193   2oc2o 7194   Nocsur 30598   bdaycbday 30600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-1o 7200  df-2o 7201  df-no 30601  df-bday 30603

This theorem is referenced by:  nobndup  30660  nobnddown  30661