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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 8854 | . 2 ⊢ (𝐴 ∈ dom card → ∃𝑟 𝑟 We 𝐴) | |
2 | weso 5105 | . . . . 5 ⊢ (𝑟 We 𝐴 → 𝑟 Or 𝐴) | |
3 | vex 3203 | . . . . 5 ⊢ 𝑟 ∈ V | |
4 | soex 7109 | . . . . 5 ⊢ ((𝑟 Or 𝐴 ∧ 𝑟 ∈ V) → 𝐴 ∈ V) | |
5 | 2, 3, 4 | sylancl 694 | . . . 4 ⊢ (𝑟 We 𝐴 → 𝐴 ∈ V) |
6 | 5 | exlimiv 1858 | . . 3 ⊢ (∃𝑟 𝑟 We 𝐴 → 𝐴 ∈ V) |
7 | unipw 4918 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
8 | weeq2 5103 | . . . . . 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 1774 | . . . 4 ⊢ (∃𝑟 𝑟 We ∪ 𝒫 𝐴 ↔ ∃𝑟 𝑟 We 𝐴) |
11 | 10 | biimpri 218 | . . 3 ⊢ (∃𝑟 𝑟 We 𝐴 → ∃𝑟 𝑟 We ∪ 𝒫 𝐴) |
12 | pwexg 4850 | . . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
13 | dfac8c 8856 | . . . . 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 8853 | . . . 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 199 | 1 ⊢ (𝐴 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∅c0 3915 𝒫 cpw 4158 ∪ cuni 4436 Or wor 5034 We wwe 5072 dom cdm 5114 ‘cfv 5888 cardccrd 8761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-wrecs 7407 df-recs 7468 df-en 7956 df-card 8765 |
This theorem is referenced by: ondomen 8860 dfac10 8959 zorn2lem7 9324 fpwwe 9468 canthnumlem 9470 canthp1lem2 9475 pwfseqlem4a 9483 pwfseqlem4 9484 fin2so 33396 |
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