Theorem supmaxlem 5678

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 3341	. . . . . . 7 |- (x = C -> (xRy <-> CRy))
2	1	notbid 673	. . . . . 6 |- (x = C -> (-. xRy <-> -. CRy))
3	2	ralbidv 2123	. . . . 5 |- (x = C -> (A.y e. B -. xRy <-> A.y e. B -. CRy))
4		breq2 3342	. . . . . . 7 |- (x = C -> (yRx <-> yRC))
5	4	imbi1d 675	. . . . . 6 |- (x = C -> ((yRx -> E.z e. B yRz) <-> (yRC -> E.z e. B yRz)))
6	5	ralbidv 2123	. . . . 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 690	. . . 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 2381	. . 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 3342	. . . . . . . 8 |- (z = y -> (CRz <-> CRy))
10	9	notbid 673	. . . . . . 7 |- (z = y -> (-. CRz <-> -. CRy))
11	10	cbvralv 2280	. . . . . 6 |- (A.z e. B -. CRz <-> A.y e. B -. CRy)
12	11	biimpi 168	. . . . 5 |- (A.z e. B -. CRz -> A.y e. B -. CRy)
13		breq2 3342	. . . . . . . . 9 |- (z = C -> (yRz <-> yRC))
14	13	rcla4ev 2381	. . . . . . . 8 |- ((C e. B /\ yRC) -> E.z e. B yRz)
15	14	ex 402	. . . . . . 7 |- (C e. B -> (yRC -> E.z e. B yRz))
16	15	a1d 15	. . . . . 6 |- (C e. B -> (y e. A -> (yRC -> E.z e. B yRz)))
17	16	r19.21aiv 2175	. . . . 5 |- (C e. B -> A.y e. A (yRC -> E.z e. B yRz))
18	12, 17	anim12i 360	. . . 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 484	. . 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 500	. 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 1063	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 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338

This theorem is referenced by:  supmax 5679

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339

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