Theorem nobndlem8 27960

Description: Lemma for nobndup 27961 and nobnddown 27962. 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 3063	. 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 27955	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) = C )
5	4, 3	syl6eq 2506	. . 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 27953	. . . . 5 |- ( F e. _V -> suc U. ( bday " F ) e. On )
7	6	adantl 466	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> suc U. ( bday " F ) e. On )
8		bdayfn 27940	. . . . . . . . . . 11 |- bday Fn No
9		fnfvima 6040	. . . . . . . . . . 11 |- ( ( bday Fn No /\ F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
10	8, 9	mp3an1 1302	. . . . . . . . . 10 |- ( ( F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
11	10	3adant2 1007	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
12		elssuni 4205	. . . . . . . . 9 |- ( ( bday ` n ) e. ( bday " F ) -> ( bday ` n ) C_ U. ( bday " F ) )
13	11, 12	syl 16	. . . . . . . 8 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) C_ U. ( bday " F ) )
14		bdayelon 27941	. . . . . . . . 9 |- ( bday ` n ) e. On
15		sucelon 6514	. . . . . . . . . . 11 |- ( U. ( bday " F ) e. On <-> suc U. ( bday " F ) e. On )
16	6, 15	sylibr 212	. . . . . . . . . 10 |- ( F e. _V -> U. ( bday " F ) e. On )
17	16	3ad2ant2 1010	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> U. ( bday " F ) e. On )
18		onsssuc 4890	. . . . . . . . 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 663	. . . . . . . 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 210	. . . . . . 7 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) e. suc U. ( bday " F ) )
21		ssel2 3435	. . . . . . . . 9 |- ( ( F C_ No /\ n e. F ) -> n e. No )
22		fvnobday 27943	. . . . . . . . . 10 |- ( n e. No -> ( n ` ( bday ` n ) ) = (/) )
23	2	nosgnn0i 27920	. . . . . . . . . . 11 |- (/) =/= S
24	23	a1i 11	. . . . . . . . . 10 |- ( n e. No -> (/) =/= S )
25	22, 24	eqnetrd 2738	. . . . . . . . 9 |- ( n e. No -> ( n ` ( bday ` n ) ) =/= S )
26	21, 25	syl 16	. . . . . . . 8 |- ( ( F C_ No /\ n e. F ) -> ( n ` ( bday ` n ) ) =/= S )
27	26	3adant2 1007	. . . . . . 7 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( n ` ( bday ` n ) ) =/= S )
28		fveq2 5775	. . . . . . . . 9 |- ( b = ( bday ` n ) -> ( n ` b ) = ( n ` ( bday ` n ) ) )
29	28	neeq1d 2722	. . . . . . . 8 |- ( b = ( bday ` n ) -> ( ( n ` b ) =/= S <-> ( n ` ( bday ` n ) ) =/= S ) )
30	29	rspcev 3155	. . . . . . 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 661	. . . . . 6 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
32	31	3expa 1188	. . . . 5 |- ( ( ( F C_ No /\ F e. _V ) /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
33	32	ralrimiva 2881	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> A. n e. F E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
34		rexeq 3000	. . . . . 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 2815	. . . . 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 4238	. . . 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 661	. . 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 3474	. 2 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) C_ suc U. ( bday " F ) )
39	1, 38	sylan2 474	1 |- ( ( F C_ No /\ F e. A ) -> ( bday ` ( C X. { S } ) ) C_ suc U. ( bday " F ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1757   =/= wne 2641   A.wral 2792   E.wrex 2793   {crab 2796   _Vcvv 3054   C_ wss 3412   (/)c0 3721   {csn 3961   {cpr 3963   U.cuni 4175   |^|cint 4212   Oncon0 4803   suc csuc 4805   X. cxp 4922   "cima 4927   Fn wfn 5497   ` cfv 5502   1oc1o 6999   2oc2o 7000   Nocsur 27901   bdaycbday 27903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-1o 7006  df-2o 7007  df-no 27904  df-bday 27906

This theorem is referenced by:  nobndup  27961  nobnddown  27962