MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  weth Unicode version

Theorem weth 8122
Description: Well-ordering theorem: any set  A can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario Carneiro, 5-Jan-2013.)
Assertion
Ref Expression
weth  |-  ( A  e.  V  ->  E. x  x  We  A )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem weth
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 weeq2 4382 . . 3  |-  ( y  =  A  ->  (
x  We  y  <->  x  We  A ) )
21exbidv 1612 . 2  |-  ( y  =  A  ->  ( E. x  x  We  y 
<->  E. x  x  We  A ) )
3 dfac8 7761 . . 3  |-  (CHOICE  <->  A. y E. x  x  We  y )
43axaci 8095 . 2  |-  E. x  x  We  y
52, 4vtoclg 2843 1  |-  ( A  e.  V  ->  E. x  x  We  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528    = wceq 1623    e. wcel 1684    We wwe 4351
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  ax-ac2 8089
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-riota 6304  df-recs 6388  df-en 6864  df-card 7572  df-ac 7743
  Copyright terms: Public domain W3C validator