Theorem supeu 4587

Description: A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general).

Hypothesis

Ref	Expression
supmo.1	|- R Or A

Assertion

Ref	Expression
supeu	|- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))

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

Proof of Theorem supeu

Step	Hyp	Ref	Expression
1		supmo.1	. . . 4 |- R Or A
2	1	supmo 4585	. . 3 |- E*x(x e. A /\ (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))
3	2	jctr 291	. 2 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) /\ E*x(x e. A /\ (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))))
4		reu5 1932	. 2 |- (E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) <-> (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) /\ E*x(x e. A /\ (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))))
5	3, 4	sylibr 200	1 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 960  E*wmo 1383  A.wral 1648  E.wrex 1649  E!wreu 1650   class class class wbr 2624   Or wor 2845

This theorem is referenced by:  supcl 4588  supub 4589  suplub 4590  supeui 4592  lbinfm 6050  infmsup 6070  supxr 6083  supxrre 6085

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-10 968  ax-11 969  ax-12 970  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  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-ral 1652  df-rex 1653  df-reu 1654  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-po 2846  df-so 2856

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