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Theorem cardlim 8783
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 3619 . . . . . . . . . . 11 ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) ↔ ω ⊆ suc 𝑥))
21biimpd 219 . . . . . . . . . 10 ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) → ω ⊆ suc 𝑥))
3 limom 7065 . . . . . . . . . . . 12 Lim ω
4 limsssuc 7035 . . . . . . . . . . . 12 (Lim ω → (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥))
53, 4ax-mp 5 . . . . . . . . . . 11 (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥)
6 infensuc 8123 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
76ex 450 . . . . . . . . . . 11 (𝑥 ∈ On → (ω ⊆ 𝑥𝑥 ≈ suc 𝑥))
85, 7syl5bir 233 . . . . . . . . . 10 (𝑥 ∈ On → (ω ⊆ suc 𝑥𝑥 ≈ suc 𝑥))
92, 8sylan9r 689 . . . . . . . . 9 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ suc 𝑥))
10 breq2 4648 . . . . . . . . . 10 ((card‘𝐴) = suc 𝑥 → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
1110adantl 482 . . . . . . . . 9 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
129, 11sylibrd 249 . . . . . . . 8 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ (card‘𝐴)))
1312ex 450 . . . . . . 7 (𝑥 ∈ On → ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) → 𝑥 ≈ (card‘𝐴))))
1413com3r 87 . . . . . 6 (ω ⊆ (card‘𝐴) → (𝑥 ∈ On → ((card‘𝐴) = suc 𝑥𝑥 ≈ (card‘𝐴))))
1514imp 445 . . . . 5 ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = suc 𝑥𝑥 ≈ (card‘𝐴)))
16 vex 3198 . . . . . . . . . 10 𝑥 ∈ V
1716sucid 5792 . . . . . . . . 9 𝑥 ∈ suc 𝑥
18 eleq2 2688 . . . . . . . . 9 ((card‘𝐴) = suc 𝑥 → (𝑥 ∈ (card‘𝐴) ↔ 𝑥 ∈ suc 𝑥))
1917, 18mpbiri 248 . . . . . . . 8 ((card‘𝐴) = suc 𝑥𝑥 ∈ (card‘𝐴))
20 cardidm 8770 . . . . . . . 8 (card‘(card‘𝐴)) = (card‘𝐴)
2119, 20syl6eleqr 2710 . . . . . . 7 ((card‘𝐴) = suc 𝑥𝑥 ∈ (card‘(card‘𝐴)))
22 cardne 8776 . . . . . . 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 187 . . . 4 ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ¬ (card‘𝐴) = suc 𝑥)
2625nrexdv 2998 . . 3 (ω ⊆ (card‘𝐴) → ¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥)
27 peano1 7070 . . . . . 6 ∅ ∈ ω
28 ssel 3589 . . . . . 6 (ω ⊆ (card‘𝐴) → (∅ ∈ ω → ∅ ∈ (card‘𝐴)))
2927, 28mpi 20 . . . . 5 (ω ⊆ (card‘𝐴) → ∅ ∈ (card‘𝐴))
30 n0i 3912 . . . . 5 (∅ ∈ (card‘𝐴) → ¬ (card‘𝐴) = ∅)
31 cardon 8755 . . . . . . . . 9 (card‘𝐴) ∈ On
3231onordi 5820 . . . . . . . 8 Ord (card‘𝐴)
33 ordzsl 7030 . . . . . . . 8 (Ord (card‘𝐴) ↔ ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
3432, 33mpbi 220 . . . . . . 7 ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴))
35 3orass 1039 . . . . . . 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 220 . . . . . 6 ((card‘𝐴) = ∅ ∨ (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
3736ori 390 . . . . 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 392 . . 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 7055 . 2 (Lim (card‘𝐴) → ω ⊆ (card‘𝐴))
4240, 41impbii 199 1 (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035   = wceq 1481  wcel 1988  wrex 2910  wss 3567  c0 3907   class class class wbr 4644  Ord word 5710  Oncon0 5711  Lim wlim 5712  suc csuc 5713  cfv 5876  ωcom 7050  cen 7937  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-1o 7545  df-er 7727  df-en 7941  df-dom 7942  df-card 8750
This theorem is referenced by:  infxpenlem  8821  alephislim  8891  cflim2  9070  winalim  9502  gruina  9625
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