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| 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 9994 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
| 2 | bren 8996 | . . 3 ⊢ ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴) | |
| 3 | 1, 2 | sylib 218 | . 2 ⊢ (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴) | 
| 4 | sqxpexg 7776 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V) | |
| 5 | incom 4208 | . . . . . 6 ⊢ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) | |
| 6 | inex1g 5318 | . . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) ∈ V) | |
| 7 | 5, 6 | eqeltrid 2844 | . . . . 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 6859 | . . . . . 6 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(card‘𝐴)) | |
| 10 | cardon 9985 | . . . . . . . 8 ⊢ (card‘𝐴) ∈ On | |
| 11 | 10 | onordi 6494 | . . . . . . 7 ⊢ Ord (card‘𝐴) | 
| 12 | ordwe 6396 | . . . . . . 7 ⊢ (Ord (card‘𝐴) → E We (card‘𝐴)) | |
| 13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ E We (card‘𝐴) | 
| 14 | eqid 2736 | . . . . . . 7 ⊢ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} = {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} | |
| 15 | 14 | f1owe 7374 | . . . . . 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 5769 | . . . . 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 218 | . . . 4 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴) | 
| 19 | weeq1 5671 | . . . . 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 3596 | . . . 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 1932 | . 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 1778 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 class class class wbr 5142 {copab 5204 E cep 5582 We wwe 5635 × cxp 5682 ◡ccnv 5683 dom cdm 5684 Ord word 6382 –1-1-onto→wf1o 6559 ‘cfv 6560 ≈ cen 8983 cardccrd 9976 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-en 8987 df-card 9980 | 
| This theorem is referenced by: ween 10076 ac5num 10077 dfac8 10177 numiunnum 36472 | 
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