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

Theorem dfac8b 10055
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 9977 . . 3 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
2 bren 8974 . . 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 7757 . . . . 5 (𝐴 ∈ dom card β†’ (𝐴 Γ— 𝐴) ∈ V)
5 incom 4201 . . . . . 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 5319 . . . . . 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 2833 . . . . 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 6851 . . . . . 6 (𝑓:(cardβ€˜π΄)–1-1-onto→𝐴 β†’ ◑𝑓:𝐴–1-1-ontoβ†’(cardβ€˜π΄))
10 cardon 9968 . . . . . . . 8 (cardβ€˜π΄) ∈ On
1110onordi 6480 . . . . . . 7 Ord (cardβ€˜π΄)
12 ordwe 6382 . . . . . . 7 (Ord (cardβ€˜π΄) β†’ E We (cardβ€˜π΄))
1311, 12ax-mp 5 . . . . . 6 E We (cardβ€˜π΄)
14 eqid 2728 . . . . . . 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 7361 . . . . . 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 5762 . . . . 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 5666 . . . . 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 3584 . . . 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 1929 . 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 1774   ∈ wcel 2099  Vcvv 3471   ∩ cin 3946   class class class wbr 5148  {copab 5210   E cep 5581   We wwe 5632   Γ— cxp 5676  β—‘ccnv 5677  dom cdm 5678  Ord word 6368  β€“1-1-ontoβ†’wf1o 6547  β€˜cfv 6548   β‰ˆ cen 8961  cardccrd 9959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-en 8965  df-card 9963
This theorem is referenced by:  ween  10059  ac5num  10060  dfac8  10159
  Copyright terms: Public domain W3C validator