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Theorem zorng 8776
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 8779 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 2874 . . . . . 6  |-  ( U. z  e.  A  <->  E. x  e.  A  x  =  U. z )
2 eqimss2 3509 . . . . . . . . 9  |-  ( x  =  U. z  ->  U. z  C_  x )
3 unissb 4223 . . . . . . . . 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 3073 . . . . . . . . . . . 12  |-  x  e. 
_V
65brrpss 6465 . . . . . . . . . . 11  |-  ( u [
C.]  x  <->  u  C.  x
)
76orbi1i 520 . . . . . . . . . 10  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  ( u  C.  x  \/  u  =  x )
)
8 sspss 3555 . . . . . . . . . 10  |-  ( u 
C_  x  <->  ( u  C.  x  \/  u  =  x ) )
97, 8bitr4i 252 . . . . . . . . 9  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  u 
C_  x )
109ralbii 2831 . . . . . . . 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 2921 . . . . . 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 1605 . . 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 6466 . . . 4  |- [ C.]  Po  A
17 zorn2g 8775 . . . 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 1303 . . 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 3073 . . . . . 6  |-  y  e. 
_V
2120brrpss 6465 . . . . 5  |-  ( x [
C.]  y  <->  x  C.  y
)
2221notbii 296 . . . 4  |-  ( -.  x [ C.]  y  <->  -.  x  C.  y )
2322ralbii 2831 . . 3  |-  ( A. y  e.  A  -.  x [ C.]  y  <->  A. y  e.  A  -.  x  C.  y )
2423rexbii 2852 . 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 1368    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796    C_ wss 3428    C. wpss 3429   U.cuni 4191   class class class wbr 4392    Po wpo 4739    Or wor 4740   dom cdm 4940   [ C.] crpss 6461   cardccrd 8208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-rpss 6462  df-recs 6934  df-en 7413  df-card 8212
This theorem is referenced by:  zornn0g  8777  zorn  8779
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