Theorem nobndlem8 30600

Description: Lemma for nobndup 30601 and nobnddown 30602. 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 3056	. 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 30595	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> ( bday ` ( C X. { S } ) ) = C )
5	4, 3	syl6eq 2503	. . 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 30593	. . . . 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 30580	. . . . . . . . . . 11 |- bday Fn No
9		fnfvima 6148	. . . . . . . . . . 11 |- ( ( bday Fn No /\ F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
10	8, 9	mp3an1 1353	. . . . . . . . . 10 |- ( ( F C_ No /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
11	10	3adant2 1028	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( bday ` n ) e. ( bday " F ) )
12		elssuni 4230	. . . . . . . . 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 30581	. . . . . . . . 9 |- ( bday ` n ) e. On
15		sucelon 6649	. . . . . . . . . . 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 1031	. . . . . . . . 9 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> U. ( bday " F ) e. On )
18		onsssuc 5513	. . . . . . . . 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 670	. . . . . . . 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 3429	. . . . . . . . 9 |- ( ( F C_ No /\ n e. F ) -> n e. No )
22		fvnobday 30583	. . . . . . . . . 10 |- ( n e. No -> ( n ` ( bday ` n ) ) = (/) )
23	2	nosgnn0i 30558	. . . . . . . . . . 11 |- (/) =/= S
24	23	a1i 11	. . . . . . . . . 10 |- ( n e. No -> (/) =/= S )
25	22, 24	eqnetrd 2693	. . . . . . . . 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 1028	. . . . . . 7 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> ( n ` ( bday ` n ) ) =/= S )
28		fveq2 5870	. . . . . . . . 9 |- ( b = ( bday ` n ) -> ( n ` b ) = ( n ` ( bday ` n ) ) )
29	28	neeq1d 2685	. . . . . . . 8 |- ( b = ( bday ` n ) -> ( ( n ` b ) =/= S <-> ( n ` ( bday ` n ) ) =/= S ) )
30	29	rspcev 3152	. . . . . . 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 667	. . . . . 6 |- ( ( F C_ No /\ F e. _V /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
32	31	3expa 1209	. . . . 5 |- ( ( ( F C_ No /\ F e. _V ) /\ n e. F ) -> E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
33	32	ralrimiva 2804	. . . 4 |- ( ( F C_ No /\ F e. _V ) -> A. n e. F E. b e. suc U. ( bday " F ) ( n ` b ) =/= S )
34		rexeq 2990	. . . . . 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 4264	. . . 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 667	. . 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 3468	. 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 986   = wceq 1446   e. wcel 1889   =/= wne 2624   A.wral 2739   E.wrex 2740   {crab 2743   _Vcvv 3047   C_ wss 3406   (/)c0 3733   {csn 3970   {cpr 3972   U.cuni 4201   |^|cint 4237   X. cxp 4835   "cima 4840   Oncon0 5426   suc csuc 5428   Fn wfn 5580   ` cfv 5585   1oc1o 7180   2oc2o 7181   Nocsur 30539   bdaycbday 30541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ord 5429  df-on 5430  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-1o 7187  df-2o 7188  df-no 30542  df-bday 30544

This theorem is referenced by:  nobndup  30601  nobnddown  30602