![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ac7 | Structured version Visualization version GIF version |
Description: An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.) |
Ref | Expression |
---|---|
ac7 | ⊢ ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ac 10150 | . 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) | |
2 | 1 | axaci 10500 | 1 ⊢ ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∃wex 1774 ⊆ wss 3947 dom cdm 5673 Fn wfn 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-ac2 10495 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ac 10150 |
This theorem is referenced by: ac7g 10506 |
Copyright terms: Public domain | W3C validator |