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Theorem cardlim 8998
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 3776 . . . . . . . . . . 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 7227 . . . . . . . . . . . 12 Lim ω
4 limsssuc 7197 . . . . . . . . . . . 12 (Lim ω → (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥))
53, 4ax-mp 5 . . . . . . . . . . 11 (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥)
6 infensuc 8294 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
76ex 397 . . . . . . . . . . 11 (𝑥 ∈ On → (ω ⊆ 𝑥𝑥 ≈ suc 𝑥))
85, 7syl5bir 233 . . . . . . . . . 10 (𝑥 ∈ On → (ω ⊆ suc 𝑥𝑥 ≈ suc 𝑥))
92, 8sylan9r 498 . . . . . . . . 9 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ suc 𝑥))
10 breq2 4790 . . . . . . . . . 10 ((card‘𝐴) = suc 𝑥 → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
1110adantl 467 . . . . . . . . 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 397 . . . . . . 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 393 . . . . 5 ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = suc 𝑥𝑥 ≈ (card‘𝐴)))
16 vex 3354 . . . . . . . . . 10 𝑥 ∈ V
1716sucid 5947 . . . . . . . . 9 𝑥 ∈ suc 𝑥
18 eleq2 2839 . . . . . . . . 9 ((card‘𝐴) = suc 𝑥 → (𝑥 ∈ (card‘𝐴) ↔ 𝑥 ∈ suc 𝑥))
1917, 18mpbiri 248 . . . . . . . 8 ((card‘𝐴) = suc 𝑥𝑥 ∈ (card‘𝐴))
20 cardidm 8985 . . . . . . . 8 (card‘(card‘𝐴)) = (card‘𝐴)
2119, 20syl6eleqr 2861 . . . . . . 7 ((card‘𝐴) = suc 𝑥𝑥 ∈ (card‘(card‘𝐴)))
22 cardne 8991 . . . . . . 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 3149 . . 3 (ω ⊆ (card‘𝐴) → ¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥)
27 peano1 7232 . . . . . 6 ∅ ∈ ω
28 ssel 3746 . . . . . 6 (ω ⊆ (card‘𝐴) → (∅ ∈ ω → ∅ ∈ (card‘𝐴)))
2927, 28mpi 20 . . . . 5 (ω ⊆ (card‘𝐴) → ∅ ∈ (card‘𝐴))
30 n0i 4068 . . . . 5 (∅ ∈ (card‘𝐴) → ¬ (card‘𝐴) = ∅)
31 cardon 8970 . . . . . . . . 9 (card‘𝐴) ∈ On
3231onordi 5975 . . . . . . . 8 Ord (card‘𝐴)
33 ordzsl 7192 . . . . . . . 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 1074 . . . . . . 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 850 . . . . 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 853 . . 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 7217 . 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  wa 382  wo 836  w3o 1070   = wceq 1631  wcel 2145  wrex 3062  wss 3723  c0 4063   class class class wbr 4786  Ord word 5865  Oncon0 5866  Lim wlim 5867  suc csuc 5868  cfv 6031  ωcom 7212  cen 8106  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-1o 7713  df-er 7896  df-en 8110  df-dom 8111  df-card 8965
This theorem is referenced by:  infxpenlem  9036  alephislim  9106  cflim2  9287  winalim  9719  gruina  9842
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