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Theorem dfac8b 9456
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 9381 . . 3 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2 bren 8517 . . 3 ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto𝐴)
31, 2sylib 220 . 2 (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto𝐴)
4 sqxpexg 7476 . . . . 5 (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5 incom 4177 . . . . . 6 ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)})
6 inex1g 5222 . . . . . 6 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}) ∈ V)
75, 6eqeltrid 2917 . . . . 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 6626 . . . . . 6 (𝑓:(card‘𝐴)–1-1-onto𝐴𝑓:𝐴1-1-onto→(card‘𝐴))
10 cardon 9372 . . . . . . . 8 (card‘𝐴) ∈ On
1110onordi 6294 . . . . . . 7 Ord (card‘𝐴)
12 ordwe 6203 . . . . . . 7 (Ord (card‘𝐴) → E We (card‘𝐴))
1311, 12ax-mp 5 . . . . . 6 E We (card‘𝐴)
14 eqid 2821 . . . . . . 7 {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}
1514f1owe 7105 . . . . . 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 5635 . . . . 5 ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
1816, 17sylib 220 . . . 4 (𝑓:(card‘𝐴)–1-1-onto𝐴 → ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
19 weeq1 5542 . . . . 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 1930 . 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 1776  wcel 2110  Vcvv 3494  cin 3934   class class class wbr 5065  {copab 5127   E cep 5463   We wwe 5512   × cxp 5552  ccnv 5553  dom cdm 5554  Ord word 6189  1-1-ontowf1o 6353  cfv 6354  cen 8505  cardccrd 9363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-en 8509  df-card 9367
This theorem is referenced by:  ween  9460  ac5num  9461  dfac8  9560
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