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

Theorem weth 9010
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 4869 . . 3 (𝑦 = 𝐴 → (𝑥 We 𝑦𝑥 We 𝐴))
21exbidv 1799 . 2 (𝑦 = 𝐴 → (∃𝑥 𝑥 We 𝑦 ↔ ∃𝑥 𝑥 We 𝐴))
3 dfac8 8650 . . 3 (CHOICE ↔ ∀𝑦𝑥 𝑥 We 𝑦)
43axaci 8983 . 2 𝑥 𝑥 We 𝑦
52, 4vtoclg 3128 1 (𝐴𝑉 → ∃𝑥 𝑥 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1468  wex 1692  wcel 1937   We wwe 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-rep 4548  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659  ax-ac2 8978
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-int 4265  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-se 4840  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-isom 5642  df-riota 6325  df-wrecs 7105  df-recs 7167  df-en 7653  df-card 8458  df-ac 8632
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator