Theorem supubt 15763

Description: Upper bound property of supremum.

Assertion

Ref	Expression
supubt	|- ((R Or A /\ E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))) -> (C e. B -> -. sup(B, A, R)RC))

Distinct variable groups:   x,A,y,z   x,B,y,z   x,C,y,z   x,R,y,z

Proof of Theorem supubt

Step	Hyp	Ref	Expression
1		raleq 2266	. . . . . 6 |- (A = if(R Or A, A, (/)) -> (A.y e. A (yRx -> E.z e. B yRz) <-> A.y e. if (R Or A, A, (/))(yRx -> E.z e. B yRz)))
2	1	anbi2d 678	. . . . 5 |- (A = if(R Or A, A, (/)) -> ((A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) <-> (A.y e. B -. xRy /\ A.y e. if (R Or A, A, (/))(yRx -> E.z e. B yRz))))
3	2	rexeqbi1dv 2272	. . . 4 |- (A = if(R Or A, A, (/)) -> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) <-> E.x e. if (R Or A, A, (/))(A.y e. B -. xRy /\ A.y e. if (R Or A, A, (/))(yRx -> E.z e. B yRz))))
4		supeq2 15760	. . . . . . 7 |- (A = if(R Or A, A, (/)) -> sup(B, A, R) = sup(B, if(R Or A, A, (/)), R))
5	4	breq1d 3348	. . . . . 6 |- (A = if(R Or A, A, (/)) -> (sup(B, A, R)RC <-> sup(B, if(R Or A, A, (/)), R)RC))
6	5	notbid 673	. . . . 5 |- (A = if(R Or A, A, (/)) -> (-. sup(B, A, R)RC <-> -. sup(B, if(R Or A, A, (/)), R)RC))
7	6	imbi2d 674	. . . 4 |- (A = if(R Or A, A, (/)) -> ((C e. B -> -. sup(B, A, R)RC) <-> (C e. B -> -. sup(B, if(R Or A, A, (/)), R)RC)))
8	3, 7	imbi12d 688	. . 3 |- (A = if(R Or A, A, (/)) -> ((E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (C e. B -> -. sup(B, A, R)RC)) <-> (E.x e. if (R Or A, A, (/))(A.y e. B -. xRy /\ A.y e. if (R Or A, A, (/))(yRx -> E.z e. B yRz)) -> (C e. B -> -. sup(B, if(R Or A, A, (/)), R)RC))))
9		soeq2 3609	. . . . 5 |- (A = if(R Or A, A, (/)) -> (R Or A <-> R Or if(R Or A, A, (/))))
10		soeq2 3609	. . . . 5 |- ((/) = if(R Or A, A, (/)) -> (R Or (/) <-> R Or if(R Or A, A, (/))))
11		so0 3621	. . . . 5 |- R Or (/)
12	9, 10, 11	elimhyp 3021	. . . 4 |- R Or if(R Or A, A, (/))
13	12	supub 5670	. . 3 |- (E.x e. if (R Or A, A, (/))(A.y e. B -. xRy /\ A.y e. if (R Or A, A, (/))(yRx -> E.z e. B yRz)) -> (C e. B -> -. sup(B, if(R Or A, A, (/)), R)RC))
14	8, 13	dedth 3011	. 2 |- (R Or A -> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (C e. B -> -. sup(B, A, R)RC)))
15	14	imp 377	1 |- ((R Or A /\ E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))) -> (C e. B -> -. sup(B, A, R)RC))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  (/)c0 2875  ifcif 2982   class class class wbr 3338   Or wor 3590  supcsup 5663

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-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-reu 2111  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-po 3591  df-so 3604  df-sup 5664

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