Theorem weth 10344

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 5603	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥 We 𝑦 ↔ 𝑥 We 𝐴))
2	1	exbidv 1923	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑥 We 𝑦 ↔ ∃𝑥 𝑥 We 𝐴))
3		dfac8 9984	. . 3 ⊢ (CHOICE ↔ ∀𝑦∃𝑥 𝑥 We 𝑦)
4	3	axaci 10317	. 2 ⊢ ∃𝑥 𝑥 We 𝑦
5	2, 4	vtoclg 3514	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 We 𝐴)

Syntax hints:   → wi 4   = wceq 1540  ∃wex 1780   ∈ wcel 2105   We wwe 5568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-ac2 10312

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-se 5570  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-isom 6482  df-riota 7286  df-ov 7332  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-en 8797  df-card 9788  df-ac 9965

This theorem is referenced by:  vitali2  44558