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

Theorem cardlim 8681
 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 (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))

Proof of Theorem cardlim
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sseq2 3590 . . . . . . . . . . 11 ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) ↔ ω ⊆ suc 𝑥))
21biimpd 218 . . . . . . . . . 10 ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) → ω ⊆ suc 𝑥))
3 limom 6972 . . . . . . . . . . . 12 Lim ω
4 limsssuc 6942 . . . . . . . . . . . 12 (Lim ω → (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥))
53, 4ax-mp 5 . . . . . . . . . . 11 (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥)
6 infensuc 8023 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
76ex 449 . . . . . . . . . . 11 (𝑥 ∈ On → (ω ⊆ 𝑥𝑥 ≈ suc 𝑥))
85, 7syl5bir 232 . . . . . . . . . 10 (𝑥 ∈ On → (ω ⊆ suc 𝑥𝑥 ≈ suc 𝑥))
92, 8sylan9r 688 . . . . . . . . 9 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ suc 𝑥))
10 breq2 4587 . . . . . . . . . 10 ((card‘𝐴) = suc 𝑥 → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
1110adantl 481 . . . . . . . . 9 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
129, 11sylibrd 248 . . . . . . . 8 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ (card‘𝐴)))
1312ex 449 . . . . . . 7 (𝑥 ∈ On → ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) → 𝑥 ≈ (card‘𝐴))))
1413com3r 85 . . . . . 6 (ω ⊆ (card‘𝐴) → (𝑥 ∈ On → ((card‘𝐴) = suc 𝑥𝑥 ≈ (card‘𝐴))))
1514imp 444 . . . . 5 ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = suc 𝑥𝑥 ≈ (card‘𝐴)))
16 vex 3176 . . . . . . . . . 10 𝑥 ∈ V
1716sucid 5721 . . . . . . . . 9 𝑥 ∈ suc 𝑥
18 eleq2 2677 . . . . . . . . 9 ((card‘𝐴) = suc 𝑥 → (𝑥 ∈ (card‘𝐴) ↔ 𝑥 ∈ suc 𝑥))
1917, 18mpbiri 247 . . . . . . . 8 ((card‘𝐴) = suc 𝑥𝑥 ∈ (card‘𝐴))
20 cardidm 8668 . . . . . . . 8 (card‘(card‘𝐴)) = (card‘𝐴)
2119, 20syl6eleqr 2699 . . . . . . 7 ((card‘𝐴) = suc 𝑥𝑥 ∈ (card‘(card‘𝐴)))
22 cardne 8674 . . . . . . 7 (𝑥 ∈ (card‘(card‘𝐴)) → ¬ 𝑥 ≈ (card‘𝐴))
2321, 22syl 17 . . . . . 6 ((card‘𝐴) = suc 𝑥 → ¬ 𝑥 ≈ (card‘𝐴))
2423a1i 11 . . . . 5 ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = suc 𝑥 → ¬ 𝑥 ≈ (card‘𝐴)))
2515, 24pm2.65d 186 . . . 4 ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ¬ (card‘𝐴) = suc 𝑥)
2625nrexdv 2984 . . 3 (ω ⊆ (card‘𝐴) → ¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥)
27 peano1 6977 . . . . . 6 ∅ ∈ ω
28 ssel 3562 . . . . . 6 (ω ⊆ (card‘𝐴) → (∅ ∈ ω → ∅ ∈ (card‘𝐴)))
2927, 28mpi 20 . . . . 5 (ω ⊆ (card‘𝐴) → ∅ ∈ (card‘𝐴))
30 n0i 3879 . . . . 5 (∅ ∈ (card‘𝐴) → ¬ (card‘𝐴) = ∅)
31 cardon 8653 . . . . . . . . 9 (card‘𝐴) ∈ On
3231onordi 5749 . . . . . . . 8 Ord (card‘𝐴)
33 ordzsl 6937 . . . . . . . 8 (Ord (card‘𝐴) ↔ ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
3432, 33mpbi 219 . . . . . . 7 ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴))
35 3orass 1034 . . . . . . 7 (((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)) ↔ ((card‘𝐴) = ∅ ∨ (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴))))
3634, 35mpbi 219 . . . . . 6 ((card‘𝐴) = ∅ ∨ (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
3736ori 389 . . . . 5 (¬ (card‘𝐴) = ∅ → (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
3829, 30, 373syl 18 . . . 4 (ω ⊆ (card‘𝐴) → (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
3938ord 391 . . 3 (ω ⊆ (card‘𝐴) → (¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 → Lim (card‘𝐴)))
4026, 39mpd 15 . 2 (ω ⊆ (card‘𝐴) → Lim (card‘𝐴))
41 limomss 6962 . 2 (Lim (card‘𝐴) → ω ⊆ (card‘𝐴))
4240, 41impbii 198 1 (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  ‘cfv 5804  ωcom 6957   ≈ 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-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-csb 3500  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-lim 5645  df-suc 5646  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-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-card 8648 This theorem is referenced by:  infxpenlem  8719  alephislim  8789  cflim2  8968  winalim  9396  gruina  9519
 Copyright terms: Public domain W3C validator