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Mirrors > Home > MPE Home > Th. List > ween | Structured version Visualization version GIF version |
Description: A set is numerable iff it can be well-ordered. (Contributed by Mario Carneiro, 5-Jan-2013.) |
Ref | Expression |
---|---|
ween | ⊢ (𝐴 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfac8b 10022 | . 2 ⊢ (𝐴 ∈ dom card → ∃𝑟 𝑟 We 𝐴) | |
2 | weso 5666 | . . . . 5 ⊢ (𝑟 We 𝐴 → 𝑟 Or 𝐴) | |
3 | vex 3479 | . . . . 5 ⊢ 𝑟 ∈ V | |
4 | soex 7907 | . . . . 5 ⊢ ((𝑟 Or 𝐴 ∧ 𝑟 ∈ V) → 𝐴 ∈ V) | |
5 | 2, 3, 4 | sylancl 587 | . . . 4 ⊢ (𝑟 We 𝐴 → 𝐴 ∈ V) |
6 | 5 | exlimiv 1934 | . . 3 ⊢ (∃𝑟 𝑟 We 𝐴 → 𝐴 ∈ V) |
7 | unipw 5449 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
8 | weeq2 5664 | . . . . . 6 ⊢ (∪ 𝒫 𝐴 = 𝐴 → (𝑟 We ∪ 𝒫 𝐴 ↔ 𝑟 We 𝐴)) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ (𝑟 We ∪ 𝒫 𝐴 ↔ 𝑟 We 𝐴) |
10 | 9 | exbii 1851 | . . . 4 ⊢ (∃𝑟 𝑟 We ∪ 𝒫 𝐴 ↔ ∃𝑟 𝑟 We 𝐴) |
11 | 10 | biimpri 227 | . . 3 ⊢ (∃𝑟 𝑟 We 𝐴 → ∃𝑟 𝑟 We ∪ 𝒫 𝐴) |
12 | pwexg 5375 | . . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
13 | dfac8c 10024 | . . . . 5 ⊢ (𝒫 𝐴 ∈ V → (∃𝑟 𝑟 We ∪ 𝒫 𝐴 → ∃𝑓∀𝑥 ∈ 𝒫 𝐴(𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝐴 ∈ V → (∃𝑟 𝑟 We ∪ 𝒫 𝐴 → ∃𝑓∀𝑥 ∈ 𝒫 𝐴(𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))) |
15 | dfac8a 10021 | . . . 4 ⊢ (𝐴 ∈ V → (∃𝑓∀𝑥 ∈ 𝒫 𝐴(𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥) → 𝐴 ∈ dom card)) | |
16 | 14, 15 | syld 47 | . . 3 ⊢ (𝐴 ∈ V → (∃𝑟 𝑟 We ∪ 𝒫 𝐴 → 𝐴 ∈ dom card)) |
17 | 6, 11, 16 | sylc 65 | . 2 ⊢ (∃𝑟 𝑟 We 𝐴 → 𝐴 ∈ dom card) |
18 | 1, 17 | impbii 208 | 1 ⊢ (𝐴 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∅c0 4321 𝒫 cpw 4601 ∪ cuni 4907 Or wor 5586 We wwe 5629 dom cdm 5675 ‘cfv 6540 cardccrd 9926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-en 8936 df-card 9930 |
This theorem is referenced by: ondomen 10028 dfac10 10128 zorn2lem7 10493 fpwwe 10637 canthnumlem 10639 canthp1lem2 10644 pwfseqlem4a 10652 pwfseqlem4 10653 fin2so 36413 |
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