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Theorem zorng 8880
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 8883 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
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 risset 2987 . . . . . 6  |-  ( U. z  e.  A  <->  E. x  e.  A  x  =  U. z )
2 eqimss2 3557 . . . . . . . . 9  |-  ( x  =  U. z  ->  U. z  C_  x )
3 unissb 4277 . . . . . . . . 9  |-  ( U. z  C_  x  <->  A. u  e.  z  u  C_  x
)
42, 3sylib 196 . . . . . . . 8  |-  ( x  =  U. z  ->  A. u  e.  z  u  C_  x )
5 vex 3116 . . . . . . . . . . . 12  |-  x  e. 
_V
65brrpss 6565 . . . . . . . . . . 11  |-  ( u [
C.]  x  <->  u  C.  x
)
76orbi1i 520 . . . . . . . . . 10  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  ( u  C.  x  \/  u  =  x )
)
8 sspss 3603 . . . . . . . . . 10  |-  ( u 
C_  x  <->  ( u  C.  x  \/  u  =  x ) )
97, 8bitr4i 252 . . . . . . . . 9  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  u 
C_  x )
109ralbii 2895 . . . . . . . 8  |-  ( A. u  e.  z  (
u [ C.]  x  \/  u  =  x )  <->  A. u  e.  z  u 
C_  x )
114, 10sylibr 212 . . . . . . 7  |-  ( x  =  U. z  ->  A. u  e.  z 
( u [ C.]  x  \/  u  =  x
) )
1211reximi 2932 . . . . . 6  |-  ( E. x  e.  A  x  =  U. z  ->  E. x  e.  A  A. u  e.  z 
( u [ C.]  x  \/  u  =  x
) )
131, 12sylbi 195 . . . . 5  |-  ( U. z  e.  A  ->  E. x  e.  A  A. u  e.  z  (
u [ C.]  x  \/  u  =  x )
)
1413imim2i 14 . . . 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 1614 . . 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 6566 . . . 4  |- [ C.]  Po  A
17 zorn2g 8879 . . . 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 1312 . . 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 474 . 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 3116 . . . . . 6  |-  y  e. 
_V
2120brrpss 6565 . . . . 5  |-  ( x [
C.]  y  <->  x  C.  y
)
2221notbii 296 . . . 4  |-  ( -.  x [ C.]  y  <->  -.  x  C.  y )
2322ralbii 2895 . . 3  |-  ( A. y  e.  A  -.  x [ C.]  y  <->  A. y  e.  A  -.  x  C.  y )
2423rexbii 2965 . 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 196 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476    C. wpss 3477   U.cuni 4245   class class class wbr 4447    Po wpo 4798    Or wor 4799   dom cdm 4999   [ C.] crpss 6561   cardccrd 8312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-rpss 6562  df-recs 7039  df-en 7514  df-card 8316
This theorem is referenced by:  zornn0g  8881  zorn  8883
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