Theorem nobndlem5 27957

Description: Lemma for nobndup 27961 and nobnddown 27962. 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 3521	. . . 4 |- { x e. On | ( A ` x ) =/= X } C_ On
2		nobndlem4.1	. . . . . . . 8 |- X e. { 1o , 2o }
3	2	nosgnn0i 27920	. . . . . . 7 |- (/) =/= X
4		fvnobday 27943	. . . . . . . 8 |- ( A e. No -> ( A ` ( bday ` A ) ) = (/) )
5	4	neeq1d 2722	. . . . . . 7 |- ( A e. No -> ( ( A ` ( bday ` A ) ) =/= X <-> (/) =/= X ) )
6	3, 5	mpbiri 233	. . . . . 6 |- ( A e. No -> ( A ` ( bday ` A ) ) =/= X )
7		bdayelon 27941	. . . . . . 7 |- ( bday ` A ) e. On
8		fveq2 5775	. . . . . . . . 9 |- ( x = ( bday ` A ) -> ( A ` x ) = ( A ` ( bday ` A ) ) )
9	8	neeq1d 2722	. . . . . . . 8 |- ( x = ( bday ` A ) -> ( ( A ` x ) =/= X <-> ( A ` ( bday ` A ) ) =/= X ) )
10	9	elrab3 3201	. . . . . . 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 3727	. . . . 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 6492	. . . 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 2983	. . . . 5 |- F/_ x { x e. On | ( A ` x ) =/= X }
18	17	nfint 4222	. . . 4 |- F/_ x |^| { x e. On | ( A ` x ) =/= X }
19		nfcv 2610	. . . 4 |- F/_ x On
20		nfcv 2610	. . . . . 6 |- F/_ x A
21	20, 18	nffv 5782	. . . . 5 |- F/_ x ( A ` |^| { x e. On | ( A ` x ) =/= X } )
22		nfcv 2610	. . . . 5 |- F/_ x X
23	21, 22	nfne 2776	. . . 4 |- F/ x ( A ` |^| { x e. On | ( A ` x ) =/= X } ) =/= X
24		fveq2 5775	. . . . 5 |- ( x = |^| { x e. On | ( A ` x ) =/= X } -> ( A ` x ) = ( A ` |^| { x e. On | ( A ` x ) =/= X } ) )
25	24	neeq1d 2722	. . . 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 3198	. . 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 1370   e. wcel 1757   =/= wne 2641   {crab 2796   C_ wss 3412   (/)c0 3721   {cpr 3963   |^|cint 4212   Oncon0 4803   ` 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