Theorem supmaxlem 4597

Description: A set that contains a greatest element satisfies the antecedent in supremum theorems. This allows sup(A, B, R) to be used in some situations without the completeness axiom. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Assertion

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

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

Proof of Theorem supmaxlem

Step	Hyp	Ref	Expression
1		breq1 2627	. . . . . . 7 |- (x = C -> (xRy <-> CRy))
2	1	negbid 613	. . . . . 6 |- (x = C -> (-. xRy <-> -. CRy))
3	2	ralbidv 1666	. . . . 5 |- (x = C -> (A.y e. B -. xRy <-> A.y e. B -. CRy))
4		breq2 2628	. . . . . . 7 |- (x = C -> (yRx <-> yRC))
5	4	imbi1d 615	. . . . . 6 |- (x = C -> ((yRx -> E.z e. B yRz) <-> (yRC -> E.z e. B yRz)))
6	5	ralbidv 1666	. . . . 5 |- (x = C -> (A.y e. A (yRx -> E.z e. B yRz) <-> A.y e. A (yRC -> E.z e. B yRz)))
7	3, 6	anbi12d 630	. . . 4 |- (x = C -> ((A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) <-> (A.y e. B -. CRy /\ A.y e. A (yRC -> E.z e. B yRz))))
8	7	rcla4ev 1880	. . 3 |- ((C e. A /\ (A.y e. B -. CRy /\ A.y e. A (yRC -> E.z e. B yRz))) -> E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))
9		breq2 2628	. . . . . . . 8 |- (z = y -> (CRz <-> CRy))
10	9	negbid 613	. . . . . . 7 |- (z = y -> (-. CRz <-> -. CRy))
11	10	cbvralv 1803	. . . . . 6 |- (A.z e. B -. CRz <-> A.y e. B -. CRy)
12	11	biimp 151	. . . . 5 |- (A.z e. B -. CRz -> A.y e. B -. CRy)
13		breq2 2628	. . . . . . . . 9 |- (z = C -> (yRz <-> yRC))
14	13	rcla4ev 1880	. . . . . . . 8 |- ((C e. B /\ yRC) -> E.z e. B yRz)
15	14	ex 373	. . . . . . 7 |- (C e. B -> (yRC -> E.z e. B yRz))
16	15	a1d 12	. . . . . 6 |- (C e. B -> (y e. A -> (yRC -> E.z e. B yRz)))
17	16	r19.21aiv 1716	. . . . 5 |- (C e. B -> A.y e. A (yRC -> E.z e. B yRz))
18	12, 17	anim12i 333	. . . 4 |- ((A.z e. B -. CRz /\ C e. B) -> (A.y e. B -. CRy /\ A.y e. A (yRC -> E.z e. B yRz)))
19	18	ancoms 438	. . 3 |- ((C e. B /\ A.z e. B -. CRz) -> (A.y e. B -. CRy /\ A.y e. A (yRC -> E.z e. B yRz)))
20	8, 19	sylan2 453	. 2 |- ((C e. A /\ (C e. B /\ A.z e. B -. CRz)) -> E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))
21	20	3impb 831	1 |- ((C e. A /\ C e. B /\ A.z e. B -. CRz) -> 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   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649   class class class wbr 2624

This theorem is referenced by:  supmax 4598

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-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-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625

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