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Theorem dfac8b 10069
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 9991 . . 3 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2 bren 8994 . . 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 7774 . . . . 5 (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5 incom 4217 . . . . . 6 ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)})
6 inex1g 5325 . . . . . 6 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}) ∈ V)
75, 6eqeltrid 2843 . . . . 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 6861 . . . . . 6 (𝑓:(card‘𝐴)–1-1-onto𝐴𝑓:𝐴1-1-onto→(card‘𝐴))
10 cardon 9982 . . . . . . . 8 (card‘𝐴) ∈ On
1110onordi 6497 . . . . . . 7 Ord (card‘𝐴)
12 ordwe 6399 . . . . . . 7 (Ord (card‘𝐴) → E We (card‘𝐴))
1311, 12ax-mp 5 . . . . . 6 E We (card‘𝐴)
14 eqid 2735 . . . . . . 7 {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}
1514f1owe 7373 . . . . . 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 5773 . . . . 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 5676 . . . . 5 (𝑥 = ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴))
2019spcegv 3597 . . . 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 1931 . 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 2106  Vcvv 3478  cin 3962   class class class wbr 5148  {copab 5210   E cep 5588   We wwe 5640   × cxp 5687  ccnv 5688  dom cdm 5689  Ord word 6385  1-1-ontowf1o 6562  cfv 6563  cen 8981  cardccrd 9973
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-en 8985  df-card 9977
This theorem is referenced by:  ween  10073  ac5num  10074  dfac8  10174  numiunnum  36453
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