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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 9445 | . 2 ⊢ (𝐴 ∈ dom card → ∃𝑟 𝑟 We 𝐴) | |
2 | weso 5539 | . . . . 5 ⊢ (𝑟 We 𝐴 → 𝑟 Or 𝐴) | |
3 | vex 3495 | . . . . 5 ⊢ 𝑟 ∈ V | |
4 | soex 7615 | . . . . 5 ⊢ ((𝑟 Or 𝐴 ∧ 𝑟 ∈ V) → 𝐴 ∈ V) | |
5 | 2, 3, 4 | sylancl 586 | . . . 4 ⊢ (𝑟 We 𝐴 → 𝐴 ∈ V) |
6 | 5 | exlimiv 1922 | . . 3 ⊢ (∃𝑟 𝑟 We 𝐴 → 𝐴 ∈ V) |
7 | unipw 5333 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
8 | weeq2 5537 | . . . . . 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 1839 | . . . 4 ⊢ (∃𝑟 𝑟 We ∪ 𝒫 𝐴 ↔ ∃𝑟 𝑟 We 𝐴) |
11 | 10 | biimpri 229 | . . 3 ⊢ (∃𝑟 𝑟 We 𝐴 → ∃𝑟 𝑟 We ∪ 𝒫 𝐴) |
12 | pwexg 5270 | . . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
13 | dfac8c 9447 | . . . . 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 9444 | . . . 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 210 | 1 ⊢ (𝐴 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 Vcvv 3492 ∅c0 4288 𝒫 cpw 4535 ∪ cuni 4830 Or wor 5466 We wwe 5506 dom cdm 5548 ‘cfv 6348 cardccrd 9352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-wrecs 7936 df-recs 7997 df-en 8498 df-card 9356 |
This theorem is referenced by: ondomen 9451 dfac10 9551 zorn2lem7 9912 fpwwe 10056 canthnumlem 10058 canthp1lem2 10063 pwfseqlem4a 10071 pwfseqlem4 10072 fin2so 34760 |
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