Theorem spwnex3 8651

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 4492	. 2 |- -. P~U.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	rabbii 1808	. . . . . 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 8649	. . . . 5 |- ((R e. U /\ A e. W) -> (R supw A) = if({x e. X | ph} =/= (/), U.{x e. X | ph}, P~U.X))
7	6	3adant3 801	. . . 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}, P~U.X))
8		rabn0 2296	. . . . . . 7 |- ({x e. X | ph} =/= (/) <-> E.x e. X ph)
9	8	negbii 187	. . . . . 6 |- (-. {x e. X | ph} =/= (/) <-> -. E.x e. X ph)
10		iffalse 2371	. . . . . 6 |- (-. {x e. X | ph} =/= (/) -> if({x e. X | ph} =/= (/), U.{x e. X | ph}, P~U.X) = P~U.X)
11	9, 10	sylbir 201	. . . . 5 |- (-. E.x e. X ph -> if({x e. X | ph} =/= (/), U.{x e. X | ph}, P~U.X) = P~U.X)
12	11	3ad2ant3 804	. . . 4 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> if({x e. X | ph} =/= (/), U.{x e. X | ph}, P~U.X) = P~U.X)
13	7, 12	eqtrd 1510	. . 3 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> (R supw A) = P~U.X)
14	13	eleq1d 1543	. 2 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> ((R supw A) e. X <-> P~U.X e. X))
15	1, 14	mtbiri 719	1 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> -. (R supw A) e. X)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  E.wrex 1649  {crab 1651  (/)c0 2283  ifcif 2365  P~cpw 2405  U.cuni 2507   class class class wbr 2624  (class class class)co 3969   sup_w cspw 8630

This theorem is referenced by:  spwnex 8657

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-opr 3971  df-oprab 3972  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376  df-spw 8636

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