Theorem spwnex3 9998

Description: When the supremum of set A doesn't exist, R sup_w A isn't in the the field of order relation R.

Hypotheses

Ref	Expression
spwval3.1	|- X = U.U.R
spwval3.2	|- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))

Assertion

Ref	Expression
spwnex3	|- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> -. (R supw A) e. X)

Distinct variable groups:   x,y,z,A   x,R,y,z   x,X,y

Proof of Theorem spwnex3

Step	Hyp	Ref	Expression
1		pwuninel 5550	. 2 |- -. ~PU.X e. X
2		spwval3.1	. . . . . 6 |- X = U.U.R
3		spwval3.2	. . . . . . . 8 |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))
4	3	a1i 8	. . . . . . 7 |- (x e. X -> (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy))))
5	4	rabbiia 2285	. . . . . 6 |- {x e. X | ph} = {x e. X | (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy))}
6	2, 5	spwval2 9996	. . . . 5 |- ((R e. U /\ A e. W) -> (R supw A) = if({x e. X | ph} =/= (/), U.{x e. X | ph}, ~PU.X))
7	6	3adant3 896	. . . 4 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> (R supw A) = if({x e. X | ph} =/= (/), U.{x e. X | ph}, ~PU.X))
8		rabn0 2893	. . . . . . 7 |- ({x e. X | ph} =/= (/) <-> E.x e. X ph)
9	8	notbii 204	. . . . . 6 |- (-. {x e. X | ph} =/= (/) <-> -. E.x e. X ph)
10		iffalse 2991	. . . . . 6 |- (-. {x e. X | ph} =/= (/) -> if({x e. X | ph} =/= (/), U.{x e. X | ph}, ~PU.X) = ~PU.X)
11	9, 10	sylbir 218	. . . . 5 |- (-. E.x e. X ph -> if({x e. X | ph} =/= (/), U.{x e. X | ph}, ~PU.X) = ~PU.X)
12	11	3ad2ant3 899	. . . 4 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> if({x e. X | ph} =/= (/), U.{x e. X | ph}, ~PU.X) = ~PU.X)
13	7, 12	eqtrd 1925	. . 3 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> (R supw A) = ~PU.X)
14	13	eleq1d 1963	. 2 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> ((R supw A) e. X <-> ~PU.X e. X))
15	1, 14	mtbiri 785	1 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> -. (R supw A) e. X)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108  (/)c0 2875  ifcif 2982  ~Pcpw 3032  U.cuni 3177   class class class wbr 3338  (class class class)co 4884   sup_w cspw 9981

This theorem is referenced by:  spwnex 10004

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-spw 9985

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