MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac8b Structured version   Visualization version   GIF version

Theorem dfac8b 8806
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 8731 . . 3 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2 bren 7916 . . 3 ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto𝐴)
31, 2sylib 208 . 2 (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto𝐴)
4 sqxpexg 6923 . . . . 5 (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V)
5 incom 3788 . . . . . 6 ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)})
6 inex1g 4766 . . . . . 6 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}) ∈ V)
75, 6syl5eqel 2702 . . . . 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 6111 . . . . . 6 (𝑓:(card‘𝐴)–1-1-onto𝐴𝑓:𝐴1-1-onto→(card‘𝐴))
10 cardon 8722 . . . . . . . 8 (card‘𝐴) ∈ On
1110onordi 5796 . . . . . . 7 Ord (card‘𝐴)
12 ordwe 5700 . . . . . . 7 (Ord (card‘𝐴) → E We (card‘𝐴))
1311, 12ax-mp 5 . . . . . 6 E We (card‘𝐴)
14 eqid 2621 . . . . . . 7 {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} = {⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)}
1514f1owe 6563 . . . . . 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 5152 . . . . 5 ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
1816, 17sylib 208 . . . 4 (𝑓:(card‘𝐴)–1-1-onto𝐴 → ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
19 weeq1 5067 . . . . 5 (𝑥 = ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑧, 𝑤⟩ ∣ (𝑓𝑧) E (𝑓𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴))
2019spcegv 3283 . . . 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 1858 . 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 1701  wcel 1987  Vcvv 3189  cin 3558   class class class wbr 4618  {copab 4677   E cep 4988   We wwe 5037   × cxp 5077  ccnv 5078  dom cdm 5079  Ord word 5686  1-1-ontowf1o 5851  cfv 5852  cen 7904  cardccrd 8713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-en 7908  df-card 8717
This theorem is referenced by:  ween  8810  ac5num  8811  dfac8  8909
  Copyright terms: Public domain W3C validator