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Theorem weth 8685
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 4730 . . 3  |-  ( y  =  A  ->  (
x  We  y  <->  x  We  A ) )
21exbidv 1680 . 2  |-  ( y  =  A  ->  ( E. x  x  We  y 
<->  E. x  x  We  A ) )
3 dfac8 8325 . . 3  |-  (CHOICE  <->  A. y E. x  x  We  y )
43axaci 8658 . 2  |-  E. x  x  We  y
52, 4vtoclg 3051 1  |-  ( A  e.  V  ->  E. x  x  We  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   E.wex 1586    e. wcel 1756    We wwe 4699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-ac2 8653
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-recs 6853  df-en 7332  df-card 8130  df-ac 8307
This theorem is referenced by: (None)
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