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Theorem dfac8b 8737
 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 8662 . . 3 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2 bren 7850 . . 3 ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto𝐴)
31, 2sylib 207 . 2 (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto𝐴)
4 sqxpexg 6861 . . . . 5 (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5 incom 3767 . . . . . 6 ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)})
6 inex1g 4729 . . . . . 6 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}) ∈ V)
75, 6syl5eqel 2692 . . . . 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 6062 . . . . . 6 (𝑓:(card‘𝐴)–1-1-onto𝐴𝑓:𝐴1-1-onto→(card‘𝐴))
10 cardon 8653 . . . . . . . 8 (card‘𝐴) ∈ On
1110onordi 5749 . . . . . . 7 Ord (card‘𝐴)
12 ordwe 5653 . . . . . . 7 (Ord (card‘𝐴) → E We (card‘𝐴))
1311, 12ax-mp 5 . . . . . 6 E We (card‘𝐴)
14 eqid 2610 . . . . . . 7 {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}
1514f1owe 6503 . . . . . 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 5109 . . . . 5 ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
1816, 17sylib 207 . . . 4 (𝑓:(card‘𝐴)–1-1-onto𝐴 → ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
19 weeq1 5026 . . . . 5 (𝑥 = ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴))
2019spcegv 3267 . . . 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 39 . . 3 (𝐴 ∈ dom card → (𝑓:(card‘𝐴)–1-1-onto𝐴 → ∃𝑥 𝑥 We 𝐴))
2221exlimdv 1848 . 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 1695   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   class class class wbr 4583  {copab 4642   E cep 4947   We wwe 4996   × cxp 5036  ◡ccnv 5037  dom cdm 5038  Ord word 5639  –1-1-onto→wf1o 5803  ‘cfv 5804   ≈ cen 7838  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-en 7842  df-card 8648 This theorem is referenced by:  ween  8741  ac5num  8742  dfac8  8840
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