Theorem zorn2 5958

Description: Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set A (with an ordering relation R) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 5950 through zorn2lem7 5956; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 5956.

Hypothesis

Ref	Expression
zorn2.1	|- A e. _V

Assertion

Ref	Expression
zorn2	|- ((R Po A /\ A.w((w C_ A /\ R Or w) -> E.x e. A A.z e. w (zRx \/ z = x))) -> E.x e. A A.y e. A -. xRy)

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

Proof of Theorem zorn2

Step	Hyp	Ref	Expression
1		zorn2.1	. 2 |- A e. _V
2		rdglem1 5145	. 2 |- {a | E.b e. On (a Fn b /\ A.c e. b (a` c) = ({<.h, k>. | k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm}}` (a |` c)))} = {d | E.f e. On (d Fn f /\ A.g e. f (d` g) = ({<.h, k>. | k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm}}` (d |` g)))}
3		eqid 1884	. 2 |- U.{a | E.b e. On (a Fn b /\ A.c e. b (a` c) = ({<.h, k>. | k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm}}` (a |` c)))} = U.{a | E.b e. On (a Fn b /\ A.c e. b (a` c) = ({<.h, k>. | k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm}}` (a |` c)))}
4		breq2 3342	. . . . 5 |- (v = r -> (sRv <-> sRr))
5	4	ralbidv 2123	. . . 4 |- (v = r -> (A.s e. ran d sRv <-> A.s e. ran d sRr))
6		breq1 3341	. . . . 5 |- (q = s -> (qRv <-> sRv))
7	6	cbvralv 2280	. . . 4 |- (A.q e. ran d qRv <-> A.s e. ran d sRv)
8	5, 7	syl5bb 591	. . 3 |- (v = r -> (A.q e. ran d qRv <-> A.s e. ran d sRr))
9	8	cbvrabv 2422	. 2 |- {v e. A | A.q e. ran d qRv} = {r e. A | A.s e. ran d sRr}
10		eqid 1884	. 2 |- {r e. A | A.s e. (U.{a | E.b e. On (a Fn b /\ A.c e. b (a` c) = ({<.h, k>. | k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm}}` (a |` c)))}"t)sRr} = {r e. A | A.s e. (U.{a | E.b e. On (a Fn b /\ A.c e. b (a` c) = ({<.h, k>. | k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm}}` (a |` c)))}"t)sRr}
11		id 73	. . . 4 |- (k = g -> k = g)
12		rneq 4186	. . . . . . . . . . . 12 |- (h = d -> ran h = ran d)
13	12	raleqdv 2269	. . . . . . . . . . 11 |- (h = d -> (A.q e. ran h qRv <-> A.q e. ran d qRv))
14	13	rabbidv 2287	. . . . . . . . . 10 |- (h = d -> {v e. A | A.q e. ran h qRv} = {v e. A | A.q e. ran d qRv})
15	14	eleq2d 1964	. . . . . . . . 9 |- (h = d -> (n e. {v e. A | A.q e. ran h qRv} <-> n e. {v e. A | A.q e. ran d qRv}))
16		raleq 2266	. . . . . . . . . . 11 |- ({v e. A | A.q e. ran h qRv} = {v e. A | A.q e. ran d qRv} -> (A.j e. {v e. A | A.q e. ran h qRv} -. jqn <-> A.j e. {v e. A | A.q e. ran d qRv} -. jqn))
17		breq1 3341	. . . . . . . . . . . . 13 |- (k = j -> (kqn <-> jqn))
18	17	notbid 673	. . . . . . . . . . . 12 |- (k = j -> (-. kqn <-> -. jqn))
19	18	cbvralv 2280	. . . . . . . . . . 11 |- (A.k e. {v e. A | A.q e. ran h qRv} -. kqn <-> A.j e. {v e. A | A.q e. ran h qRv} -. jqn)
20	16, 19	syl5bb 591	. . . . . . . . . 10 |- ({v e. A | A.q e. ran h qRv} = {v e. A | A.q e. ran d qRv} -> (A.k e. {v e. A | A.q e. ran h qRv} -. kqn <-> A.j e. {v e. A | A.q e. ran d qRv} -. jqn))
21	14, 20	syl 12	. . . . . . . . 9 |- (h = d -> (A.k e. {v e. A | A.q e. ran h qRv} -. kqn <-> A.j e. {v e. A | A.q e. ran d qRv} -. jqn))
22	15, 21	anbi12d 690	. . . . . . . 8 |- (h = d -> ((n e. {v e. A | A.q e. ran h qRv} /\ A.k e. {v e. A | A.q e. ran h qRv} -. kqn) <-> (n e. {v e. A | A.q e. ran d qRv} /\ A.j e. {v e. A | A.q e. ran d qRv} -. jqn)))
23	22	abbidv 2008	. . . . . . 7 |- (h = d -> {n | (n e. {v e. A | A.q e. ran h qRv} /\ A.k e. {v e. A | A.q e. ran h qRv} -. kqn)} = {n | (n e. {v e. A | A.q e. ran d qRv} /\ A.j e. {v e. A | A.q e. ran d qRv} -. jqn)})
24		eleq1 1957	. . . . . . . . 9 |- (m = n -> (m e. {v e. A | A.q e. ran h qRv} <-> n e. {v e. A | A.q e. ran h qRv}))
25		breq2 3342	. . . . . . . . . . 11 |- (m = n -> (kqm <-> kqn))
26	25	notbid 673	. . . . . . . . . 10 |- (m = n -> (-. kqm <-> -. kqn))
27	26	ralbidv 2123	. . . . . . . . 9 |- (m = n -> (A.k e. {v e. A | A.q e. ran h qRv} -. kqm <-> A.k e. {v e. A | A.q e. ran h qRv} -. kqn))
28	24, 27	anbi12d 690	. . . . . . . 8 |- (m = n -> ((m e. {v e. A | A.q e. ran h qRv} /\ A.k e. {v e. A | A.q e. ran h qRv} -. kqm) <-> (n e. {v e. A | A.q e. ran h qRv} /\ A.k e. {v e. A | A.q e. ran h qRv} -. kqn)))
29	28	cbvabv 2420	. . . . . . 7 |- {m | (m e. {v e. A | A.q e. ran h qRv} /\ A.k e. {v e. A | A.q e. ran h qRv} -. kqm)} = {n | (n e. {v e. A | A.q e. ran h qRv} /\ A.k e. {v e. A | A.q e. ran h qRv} -. kqn)}
30	23, 29	syl5eq 1940	. . . . . 6 |- (h = d -> {m | (m e. {v e. A | A.q e. ran h qRv} /\ A.k e. {v e. A | A.q e. ran h qRv} -. kqm)} = {n | (n e. {v e. A | A.q e. ran d qRv} /\ A.j e. {v e. A | A.q e. ran d qRv} -. jqn)})
31		df-rab 2112	. . . . . 6 |- {m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm} = {m | (m e. {v e. A | A.q e. ran h qRv} /\ A.k e. {v e. A | A.q e. ran h qRv} -. kqm)}
32		df-rab 2112	. . . . . 6 |- {n e. {v e. A | A.q e. ran d qRv} | A.j e. {v e. A | A.q e. ran d qRv} -. jqn} = {n | (n e. {v e. A | A.q e. ran d qRv} /\ A.j e. {v e. A | A.q e. ran d qRv} -. jqn)}
33	30, 31, 32	3eqtr4g 1953	. . . . 5 |- (h = d -> {m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm} = {n e. {v e. A | A.q e. ran d qRv} | A.j e. {v e. A | A.q e. ran d qRv} -. jqn})
34	33	unieqd 3188	. . . 4 |- (h = d -> U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm} = U.{n e. {v e. A | A.q e. ran d qRv} | A.j e. {v e. A | A.q e. ran d qRv} -. jqn})
35	11, 34	eqeqan12rd 1903	. . 3 |- ((h = d /\ k = g) -> (k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm} <-> g = U.{n e. {v e. A | A.q e. ran d qRv} | A.j e. {v e. A | A.q e. ran d qRv} -. jqn}))
36	35	cbvopabv 3404	. 2 |- {<.h, k>. | k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm}} = {<.d, g>. | g = U.{n e. {v e. A | A.q e. ran d qRv} | A.j e. {v e. A | A.q e. ran d qRv} -. jqn}}
37		eqid 1884	. 2 |- {r e. A | A.s e. (U.{a | E.b e. On (a Fn b /\ A.c e. b (a` c) = ({<.h, k>. | k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm}}` (a |` c)))}"u)sRr} = {r e. A | A.s e. (U.{a | E.b e. On (a Fn b /\ A.c e. b (a` c) = ({<.h, k>. | k = U.{m e. {v e. A | A.q e. ran h qRv} | A.k e. {v e. A | A.q e. ran h qRv} -. kqm}}` (a |` c)))}"u)sRr}
38	1, 2, 3, 9, 10, 36, 37	zorn2lem7 5956	1 |- ((R Po A /\ A.w((w C_ A /\ R Or w) -> E.x e. A A.z e. w (zRx \/ z = x))) -> E.x e. A A.y e. A -. xRy)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  U.cuni 3177   class class class wbr 3338  {copab 3395   Po wpo 3589   Or wor 3590  Oncon0 3657  ran crn 3987   |` cres 3988  "cima 3989   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  zorn 5959

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-13 1311  ax-14 1312  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  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-iso 4015

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