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Mirrors > Home > MPE Home > Th. List > weth | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
weth | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 We 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | weeq2 5544 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑥 We 𝑦 ↔ 𝑥 We 𝐴)) | |
2 | 1 | exbidv 1922 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑥 We 𝑦 ↔ ∃𝑥 𝑥 We 𝐴)) |
3 | dfac8 9561 | . . 3 ⊢ (CHOICE ↔ ∀𝑦∃𝑥 𝑥 We 𝑦) | |
4 | 3 | axaci 9890 | . 2 ⊢ ∃𝑥 𝑥 We 𝑦 |
5 | 2, 4 | vtoclg 3567 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1780 ∈ wcel 2114 We wwe 5513 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-ac2 9885 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-wrecs 7947 df-recs 8008 df-en 8510 df-card 9368 df-ac 9542 |
This theorem is referenced by: vitali2 42996 |
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