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Theorem cardlim 9434
 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 3918 . . . . . . . . . . 11 ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) ↔ ω ⊆ suc 𝑥))
21biimpd 232 . . . . . . . . . 10 ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) → ω ⊆ suc 𝑥))
3 limom 7594 . . . . . . . . . . . 12 Lim ω
4 limsssuc 7564 . . . . . . . . . . . 12 (Lim ω → (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥))
53, 4ax-mp 5 . . . . . . . . . . 11 (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥)
6 infensuc 8717 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
76ex 416 . . . . . . . . . . 11 (𝑥 ∈ On → (ω ⊆ 𝑥𝑥 ≈ suc 𝑥))
85, 7syl5bir 246 . . . . . . . . . 10 (𝑥 ∈ On → (ω ⊆ suc 𝑥𝑥 ≈ suc 𝑥))
92, 8sylan9r 512 . . . . . . . . 9 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ suc 𝑥))
10 breq2 5036 . . . . . . . . . 10 ((card‘𝐴) = suc 𝑥 → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
1110adantl 485 . . . . . . . . 9 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
129, 11sylibrd 262 . . . . . . . 8 ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ (card‘𝐴)))
1312ex 416 . . . . . . 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 410 . . . . 5 ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = suc 𝑥𝑥 ≈ (card‘𝐴)))
16 vex 3413 . . . . . . . . . 10 𝑥 ∈ V
1716sucid 6248 . . . . . . . . 9 𝑥 ∈ suc 𝑥
18 eleq2 2840 . . . . . . . . 9 ((card‘𝐴) = suc 𝑥 → (𝑥 ∈ (card‘𝐴) ↔ 𝑥 ∈ suc 𝑥))
1917, 18mpbiri 261 . . . . . . . 8 ((card‘𝐴) = suc 𝑥𝑥 ∈ (card‘𝐴))
20 cardidm 9421 . . . . . . . 8 (card‘(card‘𝐴)) = (card‘𝐴)
2119, 20eleqtrrdi 2863 . . . . . . 7 ((card‘𝐴) = suc 𝑥𝑥 ∈ (card‘(card‘𝐴)))
22 cardne 9427 . . . . . . 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 199 . . . 4 ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ¬ (card‘𝐴) = suc 𝑥)
2625nrexdv 3194 . . 3 (ω ⊆ (card‘𝐴) → ¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥)
27 peano1 7600 . . . . . 6 ∅ ∈ ω
28 ssel 3885 . . . . . 6 (ω ⊆ (card‘𝐴) → (∅ ∈ ω → ∅ ∈ (card‘𝐴)))
2927, 28mpi 20 . . . . 5 (ω ⊆ (card‘𝐴) → ∅ ∈ (card‘𝐴))
30 n0i 4232 . . . . 5 (∅ ∈ (card‘𝐴) → ¬ (card‘𝐴) = ∅)
31 cardon 9406 . . . . . . . . 9 (card‘𝐴) ∈ On
3231onordi 6274 . . . . . . . 8 Ord (card‘𝐴)
33 ordzsl 7559 . . . . . . . 8 (Ord (card‘𝐴) ↔ ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
3432, 33mpbi 233 . . . . . . 7 ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴))
35 3orass 1087 . . . . . . 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 233 . . . . . 6 ((card‘𝐴) = ∅ ∨ (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
3736ori 858 . . . . 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 861 . . 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 7584 . 2 (Lim (card‘𝐴) → ω ⊆ (card‘𝐴))
4240, 41impbii 212 1 (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ⊆ wss 3858  ∅c0 4225   class class class wbr 5032  Ord word 6168  Oncon0 6169  Lim wlim 6170  suc csuc 6171  ‘cfv 6335  ωcom 7579   ≈ cen 8524  cardccrd 9397 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-er 8299  df-en 8528  df-dom 8529  df-card 9401 This theorem is referenced by:  infxpenlem  9473  alephislim  9543  cflim2  9723  winalim  10155  gruina  10278
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