Theorem supeu 4578

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 4576	. . 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 1929	. 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 958  E*wmo 1381  A.wral 1645  E.wrex 1646  E!wreu 1647   class class class wbr 2619   Or wor 2839

This theorem is referenced by:  supcl 4579  supub 4580  suplub 4581  supeui 4583  lbinfm 6048  infmsup 6068  supxr 6081  supxrre 6083

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-10 966  ax-11 967  ax-12 968  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-ral 1649  df-rex 1650  df-reu 1651  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-po 2840  df-so 2850

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