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Theorem dfac8b 10072
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.)
Assertion
Ref Expression
dfac8b (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem dfac8b
Dummy variables 𝑤 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardid2 9994 . . 3 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2 bren 8996 . . 3 ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto𝐴)
31, 2sylib 218 . 2 (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto𝐴)
4 sqxpexg 7776 . . . . 5 (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5 incom 4208 . . . . . 6 ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)})
6 inex1g 5318 . . . . . 6 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}) ∈ V)
75, 6eqeltrid 2844 . . . . 5 ((𝐴 × 𝐴) ∈ V → ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) ∈ V)
84, 7syl 17 . . . 4 (𝐴 ∈ dom card → ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) ∈ V)
9 f1ocnv 6859 . . . . . 6 (𝑓:(card‘𝐴)–1-1-onto𝐴𝑓:𝐴1-1-onto→(card‘𝐴))
10 cardon 9985 . . . . . . . 8 (card‘𝐴) ∈ On
1110onordi 6494 . . . . . . 7 Ord (card‘𝐴)
12 ordwe 6396 . . . . . . 7 (Ord (card‘𝐴) → E We (card‘𝐴))
1311, 12ax-mp 5 . . . . . 6 E We (card‘𝐴)
14 eqid 2736 . . . . . . 7 {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}
1514f1owe 7374 . . . . . 6 (𝑓:𝐴1-1-onto→(card‘𝐴) → ( E We (card‘𝐴) → {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} We 𝐴))
169, 13, 15mpisyl 21 . . . . 5 (𝑓:(card‘𝐴)–1-1-onto𝐴 → {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} We 𝐴)
17 weinxp 5769 . . . . 5 ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
1816, 17sylib 218 . . . 4 (𝑓:(card‘𝐴)–1-1-onto𝐴 → ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
19 weeq1 5671 . . . . 5 (𝑥 = ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴))
2019spcegv 3596 . . . 4 (({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) ∈ V → (({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴))
218, 18, 20syl2im 40 . . 3 (𝐴 ∈ dom card → (𝑓:(card‘𝐴)–1-1-onto𝐴 → ∃𝑥 𝑥 We 𝐴))
2221exlimdv 1932 . 2 (𝐴 ∈ dom card → (∃𝑓 𝑓:(card‘𝐴)–1-1-onto𝐴 → ∃𝑥 𝑥 We 𝐴))
233, 22mpd 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-ontowf1o 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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