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Theorem zorng 8901
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 8904 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 2982 . . . . . 6  |-  ( U. z  e.  A  <->  E. x  e.  A  x  =  U. z )
2 eqimss2 3552 . . . . . . . . 9  |-  ( x  =  U. z  ->  U. z  C_  x )
3 unissb 4283 . . . . . . . . 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 3112 . . . . . . . . . . . 12  |-  x  e. 
_V
65brrpss 6582 . . . . . . . . . . 11  |-  ( u [ C.]  x  <->  u  C.  x )
76orbi1i 520 . . . . . . . . . 10  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  ( u  C.  x  \/  u  =  x )
)
8 sspss 3599 . . . . . . . . . 10  |-  ( u 
C_  x  <->  ( u  C.  x  \/  u  =  x ) )
97, 8bitr4i 252 . . . . . . . . 9  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  u 
C_  x )
109ralbii 2888 . . . . . . . 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 2925 . . . . . 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 1634 . . 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 6583 . . . 4  |- [ C.]  Po  A
17 zorn2g 8900 . . . 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 3112 . . . . . 6  |-  y  e. 
_V
2120brrpss 6582 . . . . 5  |-  ( x [ C.]  y  <->  x  C.  y )
2221notbii 296 . . . 4  |-  ( -.  x [ C.]  y  <->  -.  x  C.  y )
2322ralbii 2888 . . 3  |-  ( A. y  e.  A  -.  x [ C.]  y  <->  A. y  e.  A  -.  x  C.  y )
2423rexbii 2959 . 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 1393    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471    C. wpss 3472   U.cuni 4251   class class class wbr 4456    Po wpo 4807    Or wor 4808   dom cdm 5008   [ C.] crpss 6578   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-rpss 6579  df-recs 7060  df-en 7536  df-card 8337
This theorem is referenced by:  zornn0g  8902  zorn  8904
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