Theorem nobndlem5 29033

Description: Lemma for nobndup 29037 and nobnddown 29038. 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 3585	. . . 4 |- { x e. On | ( A ` x ) =/= X } C_ On
2		nobndlem4.1	. . . . . . . 8 |- X e. { 1o , 2o }
3	2	nosgnn0i 28996	. . . . . . 7 |- (/) =/= X
4		fvnobday 29019	. . . . . . . 8 |- ( A e. No -> ( A ` ( bday ` A ) ) = (/) )
5	4	neeq1d 2744	. . . . . . 7 |- ( A e. No -> ( ( A ` ( bday ` A ) ) =/= X <-> (/) =/= X ) )
6	3, 5	mpbiri 233	. . . . . 6 |- ( A e. No -> ( A ` ( bday ` A ) ) =/= X )
7		bdayelon 29017	. . . . . . 7 |- ( bday ` A ) e. On
8		fveq2 5864	. . . . . . . . 9 |- ( x = ( bday ` A ) -> ( A ` x ) = ( A ` ( bday ` A ) ) )
9	8	neeq1d 2744	. . . . . . . 8 |- ( x = ( bday ` A ) -> ( ( A ` x ) =/= X <-> ( A ` ( bday ` A ) ) =/= X ) )
10	9	elrab3 3262	. . . . . . 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 3791	. . . . 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 6608	. . . 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 663	. . 3 |- ( A e. No -> |^| { x e. On | ( A ` x ) =/= X } e. { x e. On | ( A ` x ) =/= X } )
17		nfrab1 3042	. . . . 5 |- F/_ x { x e. On | ( A ` x ) =/= X }
18	17	nfint 4292	. . . 4 |- F/_ x |^| { x e. On | ( A ` x ) =/= X }
19		nfcv 2629	. . . 4 |- F/_ x On
20		nfcv 2629	. . . . . 6 |- F/_ x A
21	20, 18	nffv 5871	. . . . 5 |- F/_ x ( A ` |^| { x e. On | ( A ` x ) =/= X } )
22		nfcv 2629	. . . . 5 |- F/_ x X
23	21, 22	nfne 2798	. . . 4 |- F/ x ( A ` |^| { x e. On | ( A ` x ) =/= X } ) =/= X
24		fveq2 5864	. . . . 5 |- ( x = |^| { x e. On | ( A ` x ) =/= X } -> ( A ` x ) = ( A ` |^| { x e. On | ( A ` x ) =/= X } ) )
25	24	neeq1d 2744	. . . 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 3259	. . 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 463	1 |- ( A e. No -> ( A ` |^| { x e. On | ( A ` x ) =/= X } ) =/= X )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   {crab 2818   C_ wss 3476   (/)c0 3785   {cpr 4029   |^|cint 4282   Oncon0 4878   ` cfv 5586   1oc1o 7120   2oc2o 7121   Nocsur 28977   bdaycbday 28979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-1o 7127  df-2o 7128  df-no 28980  df-bday 28982

This theorem is referenced by:  nobndup  29037  nobnddown  29038