Theorem weth 9262

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 ∈ 𝑉 → ∃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.

Step	Hyp	Ref	Expression
1		weeq2 5068	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥 We 𝑦 ↔ 𝑥 We 𝐴))
2	1	exbidv 1852	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑥 We 𝑦 ↔ ∃𝑥 𝑥 We 𝐴))
3		dfac8 8902	. . 3 ⊢ (CHOICE ↔ ∀𝑦∃𝑥 𝑥 We 𝑦)
4	3	axaci 9235	. 2 ⊢ ∃𝑥 𝑥 We 𝑦
5	2, 4	vtoclg 3257	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 We 𝐴)

Syntax hints:   → wi 4   = wceq 1480  ∃wex 1701   ∈ wcel 1992   We wwe 5037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-ac2 9230

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-wrecs 7353  df-recs 7414  df-en 7901  df-card 8710  df-ac 8884

This theorem is referenced by:  vitali2  40183