Theorem nobndlem8 30591

Description: Lemma for nobndup 30592 and nobnddown 30593. 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 3091	. 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 30586	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) = C )
5	4, 3	syl6eq 2480	. . 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 30584	. . . . 5 |- ( F e. _V -> suc U. ( bday " F ) e. On )
7	6	adantl 468	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> suc U. ( bday " F ) e. On )
8		bdayfn 30571	. . . . . . . . . . 11 |- bday Fn No
9		fnfvima 6157	. . . . . . . . . . 11 |- ( ( bday Fn No /\ F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
10	8, 9	mp3an1 1348	. . . . . . . . . 10 |- ( ( F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
11	10	3adant2 1025	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
12		elssuni 4247	. . . . . . . . 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 30572	. . . . . . . . 9 |- ( bday ` n ) e. On
15		sucelon 6657	. . . . . . . . . . 11 |- ( U. ( bday " F ) e. On <-> suc U. ( bday " F ) e. On )
16	6, 15	sylibr 216	. . . . . . . . . 10 |- ( F e. _V -> U. ( bday " F ) e. On )
17	16	3ad2ant2 1028	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> U. ( bday " F ) e. On )
18		onsssuc 5528	. . . . . . . . 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 668	. . . . . . . 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 214	. . . . . . 7 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) e. suc U. ( bday " F ) )
21		ssel2 3461	. . . . . . . . 9 |- ( ( F C_ No /\ n e. F ) -> n e. No )
22		fvnobday 30574	. . . . . . . . . 10 |- ( n e. No -> ( n ` ( bday ` n ) ) = (/) )
23	2	nosgnn0i 30551	. . . . . . . . . . 11 |- (/) =/= S
24	23	a1i 11	. . . . . . . . . 10 |- ( n e. No -> (/) =/= S )
25	22, 24	eqnetrd 2718	. . . . . . . . 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 1025	. . . . . . 7 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( n ` ( bday ` n ) ) =/= S )
28		fveq2 5880	. . . . . . . . 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 3183	. . . . . . 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 666	. . . . . 6 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
32	31	3expa 1206	. . . . 5 |- ( ( ( F C_ No /\ F e. _V ) /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
33	32	ralrimiva 2840	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> A. n e. F E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
34		rexeq 3027	. . . . . 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 2865	. . . . 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 4281	. . . 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 666	. . 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 3500	. 2 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) C_ suc U. ( bday " F ) )
39	1, 38	sylan2 477	1 |- ( ( F C_ No /\ F e. A ) -> ( bday ` ( C X. { S } ) ) C_ suc U. ( bday " F ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 983   = wceq 1438   e. wcel 1869   =/= wne 2619   A.wral 2776   E.wrex 2777   {crab 2780   _Vcvv 3082   C_ wss 3438   (/)c0 3763   {csn 3998   {cpr 4000   U.cuni 4218   |^|cint 4254   X. cxp 4850   "cima 4855   Oncon0 5441   suc csuc 5443   Fn wfn 5595   ` cfv 5600   1oc1o 7185   2oc2o 7186   Nocsur 30532   bdaycbday 30534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4535  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-pss 3454  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4219  df-int 4255  df-iun 4300  df-br 4423  df-opab 4482  df-mpt 4483  df-tr 4518  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-ord 5444  df-on 5445  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-1o 7192  df-2o 7193  df-no 30535  df-bday 30537

This theorem is referenced by:  nobndup  30592  nobnddown  30593