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

Theorem cardlim 9963
Description: An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)
Assertion
Ref Expression
cardlim (Ο‰ βŠ† (cardβ€˜π΄) ↔ Lim (cardβ€˜π΄))

Proof of Theorem cardlim
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 sseq2 4007 . . . . . . . . . . 11 ((cardβ€˜π΄) = suc π‘₯ β†’ (Ο‰ βŠ† (cardβ€˜π΄) ↔ Ο‰ βŠ† suc π‘₯))
21biimpd 228 . . . . . . . . . 10 ((cardβ€˜π΄) = suc π‘₯ β†’ (Ο‰ βŠ† (cardβ€˜π΄) β†’ Ο‰ βŠ† suc π‘₯))
3 limom 7867 . . . . . . . . . . . 12 Lim Ο‰
4 limsssuc 7835 . . . . . . . . . . . 12 (Lim Ο‰ β†’ (Ο‰ βŠ† π‘₯ ↔ Ο‰ βŠ† suc π‘₯))
53, 4ax-mp 5 . . . . . . . . . . 11 (Ο‰ βŠ† π‘₯ ↔ Ο‰ βŠ† suc π‘₯)
6 infensuc 9151 . . . . . . . . . . . 12 ((π‘₯ ∈ On ∧ Ο‰ βŠ† π‘₯) β†’ π‘₯ β‰ˆ suc π‘₯)
76ex 413 . . . . . . . . . . 11 (π‘₯ ∈ On β†’ (Ο‰ βŠ† π‘₯ β†’ π‘₯ β‰ˆ suc π‘₯))
85, 7biimtrrid 242 . . . . . . . . . 10 (π‘₯ ∈ On β†’ (Ο‰ βŠ† suc π‘₯ β†’ π‘₯ β‰ˆ suc π‘₯))
92, 8sylan9r 509 . . . . . . . . 9 ((π‘₯ ∈ On ∧ (cardβ€˜π΄) = suc π‘₯) β†’ (Ο‰ βŠ† (cardβ€˜π΄) β†’ π‘₯ β‰ˆ suc π‘₯))
10 breq2 5151 . . . . . . . . . 10 ((cardβ€˜π΄) = suc π‘₯ β†’ (π‘₯ β‰ˆ (cardβ€˜π΄) ↔ π‘₯ β‰ˆ suc π‘₯))
1110adantl 482 . . . . . . . . 9 ((π‘₯ ∈ On ∧ (cardβ€˜π΄) = suc π‘₯) β†’ (π‘₯ β‰ˆ (cardβ€˜π΄) ↔ π‘₯ β‰ˆ suc π‘₯))
129, 11sylibrd 258 . . . . . . . 8 ((π‘₯ ∈ On ∧ (cardβ€˜π΄) = suc π‘₯) β†’ (Ο‰ βŠ† (cardβ€˜π΄) β†’ π‘₯ β‰ˆ (cardβ€˜π΄)))
1312ex 413 . . . . . . 7 (π‘₯ ∈ On β†’ ((cardβ€˜π΄) = suc π‘₯ β†’ (Ο‰ βŠ† (cardβ€˜π΄) β†’ π‘₯ β‰ˆ (cardβ€˜π΄))))
1413com3r 87 . . . . . 6 (Ο‰ βŠ† (cardβ€˜π΄) β†’ (π‘₯ ∈ On β†’ ((cardβ€˜π΄) = suc π‘₯ β†’ π‘₯ β‰ˆ (cardβ€˜π΄))))
1514imp 407 . . . . 5 ((Ο‰ βŠ† (cardβ€˜π΄) ∧ π‘₯ ∈ On) β†’ ((cardβ€˜π΄) = suc π‘₯ β†’ π‘₯ β‰ˆ (cardβ€˜π΄)))
16 vex 3478 . . . . . . . . . 10 π‘₯ ∈ V
1716sucid 6443 . . . . . . . . 9 π‘₯ ∈ suc π‘₯
18 eleq2 2822 . . . . . . . . 9 ((cardβ€˜π΄) = suc π‘₯ β†’ (π‘₯ ∈ (cardβ€˜π΄) ↔ π‘₯ ∈ suc π‘₯))
1917, 18mpbiri 257 . . . . . . . 8 ((cardβ€˜π΄) = suc π‘₯ β†’ π‘₯ ∈ (cardβ€˜π΄))
20 cardidm 9950 . . . . . . . 8 (cardβ€˜(cardβ€˜π΄)) = (cardβ€˜π΄)
2119, 20eleqtrrdi 2844 . . . . . . 7 ((cardβ€˜π΄) = suc π‘₯ β†’ π‘₯ ∈ (cardβ€˜(cardβ€˜π΄)))
22 cardne 9956 . . . . . . 7 (π‘₯ ∈ (cardβ€˜(cardβ€˜π΄)) β†’ Β¬ π‘₯ β‰ˆ (cardβ€˜π΄))
2321, 22syl 17 . . . . . 6 ((cardβ€˜π΄) = suc π‘₯ β†’ Β¬ π‘₯ β‰ˆ (cardβ€˜π΄))
2423a1i 11 . . . . 5 ((Ο‰ βŠ† (cardβ€˜π΄) ∧ π‘₯ ∈ On) β†’ ((cardβ€˜π΄) = suc π‘₯ β†’ Β¬ π‘₯ β‰ˆ (cardβ€˜π΄)))
2515, 24pm2.65d 195 . . . 4 ((Ο‰ βŠ† (cardβ€˜π΄) ∧ π‘₯ ∈ On) β†’ Β¬ (cardβ€˜π΄) = suc π‘₯)
2625nrexdv 3149 . . 3 (Ο‰ βŠ† (cardβ€˜π΄) β†’ Β¬ βˆƒπ‘₯ ∈ On (cardβ€˜π΄) = suc π‘₯)
27 peano1 7875 . . . . . 6 βˆ… ∈ Ο‰
28 ssel 3974 . . . . . 6 (Ο‰ βŠ† (cardβ€˜π΄) β†’ (βˆ… ∈ Ο‰ β†’ βˆ… ∈ (cardβ€˜π΄)))
2927, 28mpi 20 . . . . 5 (Ο‰ βŠ† (cardβ€˜π΄) β†’ βˆ… ∈ (cardβ€˜π΄))
30 n0i 4332 . . . . 5 (βˆ… ∈ (cardβ€˜π΄) β†’ Β¬ (cardβ€˜π΄) = βˆ…)
31 cardon 9935 . . . . . . . . 9 (cardβ€˜π΄) ∈ On
3231onordi 6472 . . . . . . . 8 Ord (cardβ€˜π΄)
33 ordzsl 7830 . . . . . . . 8 (Ord (cardβ€˜π΄) ↔ ((cardβ€˜π΄) = βˆ… ∨ βˆƒπ‘₯ ∈ On (cardβ€˜π΄) = suc π‘₯ ∨ Lim (cardβ€˜π΄)))
3432, 33mpbi 229 . . . . . . 7 ((cardβ€˜π΄) = βˆ… ∨ βˆƒπ‘₯ ∈ On (cardβ€˜π΄) = suc π‘₯ ∨ Lim (cardβ€˜π΄))
35 3orass 1090 . . . . . . 7 (((cardβ€˜π΄) = βˆ… ∨ βˆƒπ‘₯ ∈ On (cardβ€˜π΄) = suc π‘₯ ∨ Lim (cardβ€˜π΄)) ↔ ((cardβ€˜π΄) = βˆ… ∨ (βˆƒπ‘₯ ∈ On (cardβ€˜π΄) = suc π‘₯ ∨ Lim (cardβ€˜π΄))))
3634, 35mpbi 229 . . . . . 6 ((cardβ€˜π΄) = βˆ… ∨ (βˆƒπ‘₯ ∈ On (cardβ€˜π΄) = suc π‘₯ ∨ Lim (cardβ€˜π΄)))
3736ori 859 . . . . 5 (Β¬ (cardβ€˜π΄) = βˆ… β†’ (βˆƒπ‘₯ ∈ On (cardβ€˜π΄) = suc π‘₯ ∨ Lim (cardβ€˜π΄)))
3829, 30, 373syl 18 . . . 4 (Ο‰ βŠ† (cardβ€˜π΄) β†’ (βˆƒπ‘₯ ∈ On (cardβ€˜π΄) = suc π‘₯ ∨ Lim (cardβ€˜π΄)))
3938ord 862 . . 3 (Ο‰ βŠ† (cardβ€˜π΄) β†’ (Β¬ βˆƒπ‘₯ ∈ On (cardβ€˜π΄) = suc π‘₯ β†’ Lim (cardβ€˜π΄)))
4026, 39mpd 15 . 2 (Ο‰ βŠ† (cardβ€˜π΄) β†’ Lim (cardβ€˜π΄))
41 limomss 7856 . 2 (Lim (cardβ€˜π΄) β†’ Ο‰ βŠ† (cardβ€˜π΄))
4240, 41impbii 208 1 (Ο‰ βŠ† (cardβ€˜π΄) ↔ Lim (cardβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  β€˜cfv 6540  Ο‰com 7851   β‰ˆ cen 8932  cardccrd 9926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-er 8699  df-en 8936  df-dom 8937  df-card 9930
This theorem is referenced by:  infxpenlem  10004  alephislim  10074  cflim2  10254  winalim  10686  gruina  10809
  Copyright terms: Public domain W3C validator