Theorem nobndlem5 29696

Description: Lemma for nobndup 29700 and nobnddown 29701. There is always a minimal value of a surreal that is not a given sign. (Contributed by Scott Fenton, 3-Aug-2011.)

Hypothesis

Ref	Expression
nobndlem4.1	|- X e. { 1o , 2o }

Assertion

Ref	Expression
nobndlem5	|- ( A e. No -> ( A ` |^| { x e. On | ( A ` x ) =/= X } ) =/= X )

Distinct variable groups:   x, A   x, X

Proof of Theorem nobndlem5

Step	Hyp	Ref	Expression
1		ssrab2 3571	. . . 4 |- { x e. On | ( A ` x ) =/= X } C_ On
2		nobndlem4.1	. . . . . . . 8 |- X e. { 1o , 2o }
3	2	nosgnn0i 29659	. . . . . . 7 |- (/) =/= X
4		fvnobday 29682	. . . . . . . 8 |- ( A e. No -> ( A ` ( bday ` A ) ) = (/) )
5	4	neeq1d 2731	. . . . . . 7 |- ( A e. No -> ( ( A ` ( bday ` A ) ) =/= X <-> (/) =/= X ) )
6	3, 5	mpbiri 233	. . . . . 6 |- ( A e. No -> ( A ` ( bday ` A ) ) =/= X )
7		bdayelon 29680	. . . . . . 7 |- ( bday ` A ) e. On
8		fveq2 5848	. . . . . . . . 9 |- ( x = ( bday ` A ) -> ( A ` x ) = ( A ` ( bday ` A ) ) )
9	8	neeq1d 2731	. . . . . . . 8 |- ( x = ( bday ` A ) -> ( ( A ` x ) =/= X <-> ( A ` ( bday ` A ) ) =/= X ) )
10	9	elrab3 3255	. . . . . . 7 |- ( ( bday ` A ) e. On -> ( ( bday ` A ) e. { x e. On | ( A ` x ) =/= X } <-> ( A ` ( bday ` A ) ) =/= X ) )
11	7, 10	ax-mp 5	. . . . . 6 |- ( ( bday ` A ) e. { x e. On | ( A ` x ) =/= X } <-> ( A ` ( bday ` A ) ) =/= X )
12	6, 11	sylibr 212	. . . . 5 |- ( A e. No -> ( bday ` A ) e. { x e. On | ( A ` x ) =/= X } )
13		ne0i 3789	. . . . 5 |- ( ( bday ` A ) e. { x e. On | ( A ` x ) =/= X } -> { x e. On | ( A ` x ) =/= X } =/= (/) )
14	12, 13	syl 16	. . . 4 |- ( A e. No -> { x e. On | ( A ` x ) =/= X } =/= (/) )
15		onint 6603	. . . 4 |- ( ( { x e. On | ( A ` x ) =/= X } C_ On /\ { x e. On | ( A ` x ) =/= X } =/= (/) ) -> |^| { x e. On | ( A ` x ) =/= X } e. { x e. On | ( A ` x ) =/= X } )
16	1, 14, 15	sylancr 661	. . 3 |- ( A e. No -> |^| { x e. On | ( A ` x ) =/= X } e. { x e. On | ( A ` x ) =/= X } )
17		nfrab1 3035	. . . . 5 |- F/_ x { x e. On | ( A ` x ) =/= X }
18	17	nfint 4281	. . . 4 |- F/_ x |^| { x e. On | ( A ` x ) =/= X }
19		nfcv 2616	. . . 4 |- F/_ x On
20		nfcv 2616	. . . . . 6 |- F/_ x A
21	20, 18	nffv 5855	. . . . 5 |- F/_ x ( A ` |^| { x e. On | ( A ` x ) =/= X } )
22		nfcv 2616	. . . . 5 |- F/_ x X
23	21, 22	nfne 2785	. . . 4 |- F/ x ( A ` |^| { x e. On | ( A ` x ) =/= X } ) =/= X
24		fveq2 5848	. . . . 5 |- ( x = |^| { x e. On | ( A ` x ) =/= X } -> ( A ` x ) = ( A ` |^| { x e. On | ( A ` x ) =/= X } ) )
25	24	neeq1d 2731	. . . 4 |- ( x = |^| { x e. On | ( A ` x ) =/= X } -> ( ( A ` x ) =/= X <-> ( A ` |^| { x e. On | ( A ` x ) =/= X } ) =/= X ) )
26	18, 19, 23, 25	elrabf 3252	. . 3 |- ( |^| { x e. On | ( A ` x ) =/= X } e. { x e. On | ( A ` x ) =/= X } <-> ( |^| { x e. On | ( A ` x ) =/= X } e. On /\ ( A ` |^| { x e. On | ( A ` x ) =/= X } ) =/= X ) )
27	16, 26	sylib 196	. 2 |- ( A e. No -> ( |^| { x e. On | ( A ` x ) =/= X } e. On /\ ( A ` |^| { x e. On | ( A ` x ) =/= X } ) =/= X ) )
28	27	simprd 461	1 |- ( A e. No -> ( A ` |^| { x e. On | ( A ` x ) =/= X } ) =/= X )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   {crab 2808   C_ wss 3461   (/)c0 3783   {cpr 4018   |^|cint 4271   Oncon0 4867   ` cfv 5570   1oc1o 7115   2oc2o 7116   Nocsur 29640   bdaycbday 29642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-1o 7122  df-2o 7123  df-no 29643  df-bday 29645

This theorem is referenced by:  nobndup  29700  nobnddown  29701