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Theorem zorng 8131
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 8134 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 2590 . . . . . 6  |-  ( U. z  e.  A  <->  E. x  e.  A  x  =  U. z )
2 eqimss2 3231 . . . . . . . . 9  |-  ( x  =  U. z  ->  U. z  C_  x )
3 unissb 3857 . . . . . . . . 9  |-  ( U. z  C_  x  <->  A. u  e.  z  u  C_  x
)
42, 3sylib 188 . . . . . . . 8  |-  ( x  =  U. z  ->  A. u  e.  z  u  C_  x )
5 vex 2791 . . . . . . . . . . . 12  |-  x  e. 
_V
65brrpss 6280 . . . . . . . . . . 11  |-  ( u [
C.]  x  <->  u  C.  x )
76orbi1i 506 . . . . . . . . . 10  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  ( u  C.  x  \/  u  =  x ) )
8 sspss 3275 . . . . . . . . . 10  |-  ( u 
C_  x  <->  ( u  C.  x  \/  u  =  x ) )
97, 8bitr4i 243 . . . . . . . . 9  |-  ( ( u [ C.]  x  \/  u  =  x )  <->  u 
C_  x )
109ralbii 2567 . . . . . . . 8  |-  ( A. u  e.  z  (
u [ C.]  x  \/  u  =  x )  <->  A. u  e.  z  u 
C_  x )
114, 10sylibr 203 . . . . . . 7  |-  ( x  =  U. z  ->  A. u  e.  z 
( u [ C.]  x  \/  u  =  x
) )
1211reximi 2650 . . . . . 6  |-  ( E. x  e.  A  x  =  U. z  ->  E. x  e.  A  A. u  e.  z 
( u [ C.]  x  \/  u  =  x
) )
131, 12sylbi 187 . . . . 5  |-  ( U. z  e.  A  ->  E. x  e.  A  A. u  e.  z  (
u [ C.]  x  \/  u  =  x )
)
1413imim2i 13 . . . 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 6281 . . . 4  |- [ C.]  Po  A
17 zorn2g 8130 . . . 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 1265 . . 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 460 . 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 2791 . . . . . 6  |-  y  e. 
_V
2120brrpss 6280 . . . . 5  |-  ( x [
C.]  y  <->  x  C.  y )
2221notbii 287 . . . 4  |-  ( -.  x [ C.]  y  <->  -.  x  C.  y )
2322ralbii 2567 . . 3  |-  ( A. y  e.  A  -.  x [ C.]  y  <->  A. y  e.  A  -.  x  C.  y )
2423rexbii 2568 . 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 188 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 3    -> wi 4    \/ wo 357    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152    C. wpss 3153   U.cuni 3827   class class class wbr 4023    Po wpo 4312    Or wor 4313   dom cdm 4689   [ C.] crpss 6276   cardccrd 7568
This theorem is referenced by:  zornn0g  8132  zorn  8134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-rpss 6277  df-riota 6304  df-recs 6388  df-en 6864  df-card 7572
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