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

Theorem weth 10535
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 (𝐴𝑉 → ∃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 5673 . . 3 (𝑦 = 𝐴 → (𝑥 We 𝑦𝑥 We 𝐴))
21exbidv 1921 . 2 (𝑦 = 𝐴 → (∃𝑥 𝑥 We 𝑦 ↔ ∃𝑥 𝑥 We 𝐴))
3 dfac8 10176 . . 3 (CHOICE ↔ ∀𝑦𝑥 𝑥 We 𝑦)
43axaci 10508 . 2 𝑥 𝑥 We 𝑦
52, 4vtoclg 3554 1 (𝐴𝑉 → ∃𝑥 𝑥 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wex 1779  wcel 2108   We wwe 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-en 8986  df-card 9979  df-ac 10156
This theorem is referenced by:  vitali2  46709
  Copyright terms: Public domain W3C validator