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

Theorem dfac8b 9972
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 β†’ βˆƒπ‘₯ π‘₯ We 𝐴)
Distinct variable group:   π‘₯,𝐴

Proof of Theorem dfac8b
Dummy variables 𝑀 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardid2 9894 . . 3 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
2 bren 8896 . . 3 ((cardβ€˜π΄) β‰ˆ 𝐴 ↔ βˆƒπ‘“ 𝑓:(cardβ€˜π΄)–1-1-onto→𝐴)
31, 2sylib 217 . 2 (𝐴 ∈ dom card β†’ βˆƒπ‘“ 𝑓:(cardβ€˜π΄)–1-1-onto→𝐴)
4 sqxpexg 7690 . . . . 5 (𝐴 ∈ dom card β†’ (𝐴 Γ— 𝐴) ∈ V)
5 incom 4162 . . . . . 6 ({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} ∩ (𝐴 Γ— 𝐴)) = ((𝐴 Γ— 𝐴) ∩ {βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)})
6 inex1g 5277 . . . . . 6 ((𝐴 Γ— 𝐴) ∈ V β†’ ((𝐴 Γ— 𝐴) ∩ {βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)}) ∈ V)
75, 6eqeltrid 2838 . . . . 5 ((𝐴 Γ— 𝐴) ∈ V β†’ ({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} ∩ (𝐴 Γ— 𝐴)) ∈ V)
84, 7syl 17 . . . 4 (𝐴 ∈ dom card β†’ ({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} ∩ (𝐴 Γ— 𝐴)) ∈ V)
9 f1ocnv 6797 . . . . . 6 (𝑓:(cardβ€˜π΄)–1-1-onto→𝐴 β†’ ◑𝑓:𝐴–1-1-ontoβ†’(cardβ€˜π΄))
10 cardon 9885 . . . . . . . 8 (cardβ€˜π΄) ∈ On
1110onordi 6429 . . . . . . 7 Ord (cardβ€˜π΄)
12 ordwe 6331 . . . . . . 7 (Ord (cardβ€˜π΄) β†’ E We (cardβ€˜π΄))
1311, 12ax-mp 5 . . . . . 6 E We (cardβ€˜π΄)
14 eqid 2733 . . . . . . 7 {βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} = {βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)}
1514f1owe 7299 . . . . . 6 (◑𝑓:𝐴–1-1-ontoβ†’(cardβ€˜π΄) β†’ ( E We (cardβ€˜π΄) β†’ {βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} We 𝐴))
169, 13, 15mpisyl 21 . . . . 5 (𝑓:(cardβ€˜π΄)–1-1-onto→𝐴 β†’ {βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} We 𝐴)
17 weinxp 5717 . . . . 5 ({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} We 𝐴 ↔ ({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} ∩ (𝐴 Γ— 𝐴)) We 𝐴)
1816, 17sylib 217 . . . 4 (𝑓:(cardβ€˜π΄)–1-1-onto→𝐴 β†’ ({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} ∩ (𝐴 Γ— 𝐴)) We 𝐴)
19 weeq1 5622 . . . . 5 (π‘₯ = ({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} ∩ (𝐴 Γ— 𝐴)) β†’ (π‘₯ We 𝐴 ↔ ({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} ∩ (𝐴 Γ— 𝐴)) We 𝐴))
2019spcegv 3555 . . . 4 (({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} ∩ (𝐴 Γ— 𝐴)) ∈ V β†’ (({βŸ¨π‘§, π‘€βŸ© ∣ (β—‘π‘“β€˜π‘§) E (β—‘π‘“β€˜π‘€)} ∩ (𝐴 Γ— 𝐴)) We 𝐴 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
218, 18, 20syl2im 40 . . 3 (𝐴 ∈ dom card β†’ (𝑓:(cardβ€˜π΄)–1-1-onto→𝐴 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
2221exlimdv 1937 . 2 (𝐴 ∈ dom card β†’ (βˆƒπ‘“ 𝑓:(cardβ€˜π΄)–1-1-onto→𝐴 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
233, 22mpd 15 1 (𝐴 ∈ dom card β†’ βˆƒπ‘₯ π‘₯ We 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3444   ∩ cin 3910   class class class wbr 5106  {copab 5168   E cep 5537   We wwe 5588   Γ— cxp 5632  β—‘ccnv 5633  dom cdm 5634  Ord word 6317  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497   β‰ˆ cen 8883  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-en 8887  df-card 9880
This theorem is referenced by:  ween  9976  ac5num  9977  dfac8  10076
  Copyright terms: Public domain W3C validator