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Theorem zorng 8085
Description: Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 8088 avoids the Axiom of Choice by assuming that  A is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
zorng  |-  ( ( A  e.  dom  card  /\ 
A. z ( ( z  C_  A  /\ [ C.] 
Or  z )  ->  U. z  e.  A
) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
Distinct variable group:    x, y, z, A

Proof of Theorem zorng
StepHypRef Expression
1 risset 2563 . . . . . 6  |-  ( U. z  e.  A  <->  E. x  e.  A  x  =  U. z )
2 eqimss2 3192 . . . . . . . . 9  |-  ( x  =  U. z  ->  U. z  C_  x )
3 unissb 3817 . . . . . . . . 9  |-  ( U. z  C_  x  <->  A. u  e.  z  u  C_  x
)
42, 3sylib 190 . . . . . . . 8  |-  ( x  =  U. z  ->  A. u  e.  z  u  C_  x )
5 vex 2760 . . . . . . . . . . . 12  |-  x  e. 
_V
65brrpss 6200 . . . . . . . . . . 11  |-  ( u [
C.]  x  <->  u  C.  x )
76orbi1i 508 . . . . . . . . . 10  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  ( u  C.  x  \/  u  =  x ) )
8 sspss 3236 . . . . . . . . . 10  |-  ( u 
C_  x  <->  ( u  C.  x  \/  u  =  x ) )
97, 8bitr4i 245 . . . . . . . . 9  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  u 
C_  x )
109ralbii 2540 . . . . . . . 8  |-  ( A. u  e.  z  (
u [ C.]  x  \/  u  =  x )  <->  A. u  e.  z  u 
C_  x )
114, 10sylibr 205 . . . . . . 7  |-  ( x  =  U. z  ->  A. u  e.  z 
( u [ C.]  x  \/  u  =  x
) )
1211reximi 2623 . . . . . 6  |-  ( E. x  e.  A  x  =  U. z  ->  E. x  e.  A  A. u  e.  z 
( u [ C.]  x  \/  u  =  x
) )
131, 12sylbi 189 . . . . 5  |-  ( U. z  e.  A  ->  E. x  e.  A  A. u  e.  z  (
u [ C.]  x  \/  u  =  x )
)
1413imim2i 15 . . . 4  |-  ( ( ( z  C_  A  /\ [ C.]  Or  z )  ->  U. z  e.  A
)  ->  ( (
z  C_  A  /\ [ C.] 
Or  z )  ->  E. x  e.  A  A. u  e.  z 
( u [ C.]  x  \/  u  =  x
) ) )
1514alimi 1546 . . 3  |-  ( A. z ( ( z 
C_  A  /\ [ C.]  Or  z )  ->  U. z  e.  A )  ->  A. z
( ( z  C_  A  /\ [ C.]  Or  z
)  ->  E. x  e.  A  A. u  e.  z  ( u [ C.]  x  \/  u  =  x ) ) )
16 porpss 6201 . . . 4  |- [ C.]  Po  A
17 zorn2g 8084 . . . 4  |-  ( ( A  e.  dom  card  /\ [
C.]  Po  A  /\  A. z ( ( z 
C_  A  /\ [ C.]  Or  z )  ->  E. x  e.  A  A. u  e.  z  ( u [ C.]  x  \/  u  =  x ) ) )  ->  E. x  e.  A  A. y  e.  A  -.  x [ C.]  y )
1816, 17mp3an2 1270 . . 3  |-  ( ( A  e.  dom  card  /\ 
A. z ( ( z  C_  A  /\ [ C.] 
Or  z )  ->  E. x  e.  A  A. u  e.  z 
( u [ C.]  x  \/  u  =  x
) ) )  ->  E. x  e.  A  A. y  e.  A  -.  x [ C.]  y )
1915, 18sylan2 462 . 2  |-  ( ( A  e.  dom  card  /\ 
A. z ( ( z  C_  A  /\ [ C.] 
Or  z )  ->  U. z  e.  A
) )  ->  E. x  e.  A  A. y  e.  A  -.  x [ C.]  y )
20 vex 2760 . . . . . 6  |-  y  e. 
_V
2120brrpss 6200 . . . . 5  |-  ( x [
C.]  y  <->  x  C.  y )
2221notbii 289 . . . 4  |-  ( -.  x [ C.]  y  <->  -.  x  C.  y )
2322ralbii 2540 . . 3  |-  ( A. y  e.  A  -.  x [ C.]  y  <->  A. y  e.  A  -.  x  C.  y )
2423rexbii 2541 . 2  |-  ( E. x  e.  A  A. y  e.  A  -.  x [ C.]  y  <->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
2519, 24sylib 190 1  |-  ( ( A  e.  dom  card  /\ 
A. z ( ( z  C_  A  /\ [ C.] 
Or  z )  ->  U. z  e.  A
) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    \/ wo 359    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517    C_ wss 3113    C. wpss 3114   U.cuni 3787   class class class wbr 3983    Po wpo 4270    Or wor 4271   dom cdm 4647   [ C.] crpss 6196   cardccrd 7522
This theorem is referenced by:  zornn0g  8086  zorn  8088
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-rpss 6197  df-iota 6211  df-riota 6258  df-recs 6342  df-en 6818  df-card 7526
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