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Theorem winalim 9555
Description: A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
winalim (𝐴 ∈ Inaccw → Lim 𝐴)

Proof of Theorem winalim
StepHypRef Expression
1 winainf 9554 . 2 (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
2 winacard 9552 . . 3 (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
3 cardlim 8836 . . . 4 (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))
4 sseq2 3660 . . . . 5 ((card‘𝐴) = 𝐴 → (ω ⊆ (card‘𝐴) ↔ ω ⊆ 𝐴))
5 limeq 5773 . . . . 5 ((card‘𝐴) = 𝐴 → (Lim (card‘𝐴) ↔ Lim 𝐴))
64, 5bibi12d 334 . . . 4 ((card‘𝐴) = 𝐴 → ((ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴)) ↔ (ω ⊆ 𝐴 ↔ Lim 𝐴)))
73, 6mpbii 223 . . 3 ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴 ↔ Lim 𝐴))
82, 7syl 17 . 2 (𝐴 ∈ Inaccw → (ω ⊆ 𝐴 ↔ Lim 𝐴))
91, 8mpbid 222 1 (𝐴 ∈ Inaccw → Lim 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wss 3607  Lim wlim 5762  cfv 5926  ωcom 7107  cardccrd 8799  Inaccwcwina 9542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803  df-cf 8805  df-wina 9544
This theorem is referenced by:  inar1  9635  inatsk  9638  tskuni  9643  grur1a  9679
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