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Theorem weth 8788
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 4782 . . 3  |-  ( y  =  A  ->  (
x  We  y  <->  x  We  A ) )
21exbidv 1722 . 2  |-  ( y  =  A  ->  ( E. x  x  We  y 
<->  E. x  x  We  A ) )
3 dfac8 8428 . . 3  |-  (CHOICE  <->  A. y E. x  x  We  y )
43axaci 8761 . 2  |-  E. x  x  We  y
52, 4vtoclg 3092 1  |-  ( A  e.  V  ->  E. x  x  We  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399   E.wex 1620    e. wcel 1826    We wwe 4751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-ac2 8756
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-recs 6960  df-en 7436  df-card 8233  df-ac 8410
This theorem is referenced by: (None)
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