Theorem spwnex3 8655

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 4486	. 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 1805	. . . . . 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 8653	. . . . 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 799	. . . 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 2292	. . . . . . 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 2367	. . . . . 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 802	. . . 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 1507	. . 3 |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> (R supw A) = P~U.X)
14	13	eleq1d 1540	. 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 717	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 775   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646  {crab 1648  (/)c0 2280  ifcif 2361  P~cpw 2401  U.cuni 2503   class class class wbr 2619  (class class class)co 3963   sup_w cspw 8634

This theorem is referenced by:  spwnex 8661

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-spw 8640

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