Theorem nobndlem8 29624

Description: Lemma for nobndup 29625 and nobnddown 29626. 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 3043	. 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 29619	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) = C )
5	4, 3	syl6eq 2439	. . 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 29617	. . . . 5 |- ( F e. _V -> suc U. ( bday " F ) e. On )
7	6	adantl 464	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> suc U. ( bday " F ) e. On )
8		bdayfn 29604	. . . . . . . . . . 11 |- bday Fn No
9		fnfvima 6051	. . . . . . . . . . 11 |- ( ( bday Fn No /\ F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
10	8, 9	mp3an1 1309	. . . . . . . . . 10 |- ( ( F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
11	10	3adant2 1013	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
12		elssuni 4192	. . . . . . . . 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 29605	. . . . . . . . 9 |- ( bday ` n ) e. On
15		sucelon 6551	. . . . . . . . . . 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 1016	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> U. ( bday " F ) e. On )
18		onsssuc 4879	. . . . . . . . 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 661	. . . . . . . 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 3412	. . . . . . . . 9 |- ( ( F C_ No /\ n e. F ) -> n e. No )
22		fvnobday 29607	. . . . . . . . . 10 |- ( n e. No -> ( n ` ( bday ` n ) ) = (/) )
23	2	nosgnn0i 29584	. . . . . . . . . . 11 |- (/) =/= S
24	23	a1i 11	. . . . . . . . . 10 |- ( n e. No -> (/) =/= S )
25	22, 24	eqnetrd 2675	. . . . . . . . 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 1013	. . . . . . 7 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( n ` ( bday ` n ) ) =/= S )
28		fveq2 5774	. . . . . . . . 9 |- ( b = ( bday ` n ) -> ( n ` b ) = ( n ` ( bday ` n ) ) )
29	28	neeq1d 2659	. . . . . . . 8 |- ( b = ( bday ` n ) -> ( ( n ` b ) =/= S <-> ( n ` ( bday ` n ) ) =/= S ) )
30	29	rspcev 3135	. . . . . . 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 659	. . . . . 6 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
32	31	3expa 1194	. . . . 5 |- ( ( ( F C_ No /\ F e. _V ) /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
33	32	ralrimiva 2796	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> A. n e. F E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
34		rexeq 2980	. . . . . 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 2821	. . . . 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 4226	. . . 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 659	. . 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 3451	. 2 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) C_ suc U. ( bday " F ) )
39	1, 38	sylan2 472	1 |- ( ( F C_ No /\ F e. A ) -> ( bday ` ( C X. { S } ) ) C_ suc U. ( bday " F ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1399   e. wcel 1826   =/= wne 2577   A.wral 2732   E.wrex 2733   {crab 2736   _Vcvv 3034   C_ wss 3389   (/)c0 3711   {csn 3944   {cpr 3946   U.cuni 4163   |^|cint 4199   Oncon0 4792   suc csuc 4794   X. cxp 4911   "cima 4916   Fn wfn 5491   ` cfv 5496   1oc1o 7041   2oc2o 7042   Nocsur 29565   bdaycbday 29567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-1o 7048  df-2o 7049  df-no 29568  df-bday 29570

This theorem is referenced by:  nobndup  29625  nobnddown  29626