Theorem supnub 4582

Description: An upper bound is not less than the supremum.

Hypothesis

Ref	Expression
supmo.1	|- R Or A

Assertion

Ref	Expression
supnub	|- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> ((C e. A /\ A.z e. B -. CRz) -> -. CRsup(B, A, R)))

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

Proof of Theorem supnub

Step	Hyp	Ref	Expression
1		supmo.1	. . . . . . . 8 |- R Or A
2	1	suplub 4581	. . . . . . 7 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> ((C e. A /\ CRsup(B, A, R)) -> E.z e. B CRz))
3	2	exp3a 375	. . . . . 6 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (C e. A -> (CRsup(B, A, R) -> E.z e. B CRz)))
4	3	imp 350	. . . . 5 |- ((E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) /\ C e. A) -> (CRsup(B, A, R) -> E.z e. B CRz))
5		dfrex2 1656	. . . . 5 |- (E.z e. B CRz <-> -. A.z e. B -. CRz)
6	4, 5	syl6ib 212	. . . 4 |- ((E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) /\ C e. A) -> (CRsup(B, A, R) -> -. A.z e. B -. CRz))
7	6	con2d 91	. . 3 |- ((E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) /\ C e. A) -> (A.z e. B -. CRz -> -. CRsup(B, A, R)))
8	7	ex 373	. 2 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (C e. A -> (A.z e. B -. CRz -> -. CRsup(B, A, R))))
9	8	imp3a 361	1 |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> ((C e. A /\ A.z e. B -. CRz) -> -. CRsup(B, A, R)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 958  A.wral 1645  E.wrex 1646   class class class wbr 2619   Or wor 2839  supcsup 4573

This theorem is referenced by:  supnubi 4587  supmax 4589  suprnub 6058  supxrleub 6099

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-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-sep 2703  ax-pow 2742  ax-un 2866

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-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-po 2840  df-so 2850  df-sup 4574

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