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Mirrors > Home > MPE Home > Th. List > dfac8b | Structured version Visualization version GIF version |
Description: The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
dfac8b | ⊢ (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardid2 9092 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
2 | bren 8231 | . . 3 ⊢ ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴) | |
3 | 1, 2 | sylib 210 | . 2 ⊢ (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴) |
4 | sqxpexg 7224 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V) | |
5 | incom 4032 | . . . . . 6 ⊢ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) | |
6 | inex1g 5026 | . . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) ∈ V) | |
7 | 5, 6 | syl5eqel 2910 | . . . . 5 ⊢ ((𝐴 × 𝐴) ∈ V → ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ (𝐴 ∈ dom card → ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V) |
9 | f1ocnv 6390 | . . . . . 6 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(card‘𝐴)) | |
10 | cardon 9083 | . . . . . . . 8 ⊢ (card‘𝐴) ∈ On | |
11 | 10 | onordi 6067 | . . . . . . 7 ⊢ Ord (card‘𝐴) |
12 | ordwe 5976 | . . . . . . 7 ⊢ (Ord (card‘𝐴) → E We (card‘𝐴)) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ E We (card‘𝐴) |
14 | eqid 2825 | . . . . . . 7 ⊢ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} = {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} | |
15 | 14 | f1owe 6858 | . . . . . 6 ⊢ (◡𝑓:𝐴–1-1-onto→(card‘𝐴) → ( E We (card‘𝐴) → {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴)) |
16 | 9, 13, 15 | mpisyl 21 | . . . . 5 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴) |
17 | weinxp 5421 | . . . . 5 ⊢ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴 ↔ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴) | |
18 | 16, 17 | sylib 210 | . . . 4 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴) |
19 | weeq1 5330 | . . . . 5 ⊢ (𝑥 = ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)) | |
20 | 19 | spcegv 3511 | . . . 4 ⊢ (({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V → (({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴)) |
21 | 8, 18, 20 | syl2im 40 | . . 3 ⊢ (𝐴 ∈ dom card → (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ∃𝑥 𝑥 We 𝐴)) |
22 | 21 | exlimdv 2034 | . 2 ⊢ (𝐴 ∈ dom card → (∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴 → ∃𝑥 𝑥 We 𝐴)) |
23 | 3, 22 | mpd 15 | 1 ⊢ (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1880 ∈ wcel 2166 Vcvv 3414 ∩ cin 3797 class class class wbr 4873 {copab 4935 E cep 5254 We wwe 5300 × cxp 5340 ◡ccnv 5341 dom cdm 5342 Ord word 5962 –1-1-onto→wf1o 6122 ‘cfv 6123 ≈ cen 8219 cardccrd 9074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-en 8223 df-card 9078 |
This theorem is referenced by: ween 9171 ac5num 9172 dfac8 9272 |
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