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Theorem alephcard 8878
Description: Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephcard (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)

Proof of Theorem alephcard
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6178 . . . . 5 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
21fveq2d 6182 . . . 4 (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
32, 1eqeq12d 2635 . . 3 (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4 fveq2 6178 . . . . 5 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
54fveq2d 6182 . . . 4 (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
65, 4eqeq12d 2635 . . 3 (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7 fveq2 6178 . . . . 5 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
87fveq2d 6182 . . . 4 (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
98, 7eqeq12d 2635 . . 3 (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10 fveq2 6178 . . . . 5 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
1110fveq2d 6182 . . . 4 (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
1211, 10eqeq12d 2635 . . 3 (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13 cardom 8797 . . . 4 (card‘ω) = ω
14 aleph0 8874 . . . . 5 (ℵ‘∅) = ω
1514fveq2i 6181 . . . 4 (card‘(ℵ‘∅)) = (card‘ω)
1613, 15, 143eqtr4i 2652 . . 3 (card‘(ℵ‘∅)) = (ℵ‘∅)
17 harcard 8789 . . . . 5 (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18 alephsuc 8876 . . . . . 6 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
1918fveq2d 6182 . . . . 5 (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
2017, 19, 183eqtr4a 2680 . . . 4 (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦))
2120a1d 25 . . 3 (𝑦 ∈ On → ((card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
22 vex 3198 . . . . . . 7 𝑥 ∈ V
23 cardiun 8793 . . . . . . 7 (𝑥 ∈ V → (∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘ 𝑦𝑥 (ℵ‘𝑦)) = 𝑦𝑥 (ℵ‘𝑦)))
2422, 23ax-mp 5 . . . . . 6 (∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘ 𝑦𝑥 (ℵ‘𝑦)) = 𝑦𝑥 (ℵ‘𝑦))
2524adantl 482 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘ 𝑦𝑥 (ℵ‘𝑦)) = 𝑦𝑥 (ℵ‘𝑦))
26 alephlim 8875 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2722, 26mpan 705 . . . . . . 7 (Lim 𝑥 → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2827adantr 481 . . . . . 6 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2928fveq2d 6182 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘ 𝑦𝑥 (ℵ‘𝑦)))
3025, 29, 283eqtr4d 2664 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥))
3130ex 450 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)))
323, 6, 9, 12, 16, 21, 31tfinds 7044 . 2 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33 card0 8769 . . 3 (card‘∅) = ∅
34 alephfnon 8873 . . . . . . 7 ℵ Fn On
35 fndm 5978 . . . . . . 7 (ℵ Fn On → dom ℵ = On)
3634, 35ax-mp 5 . . . . . 6 dom ℵ = On
3736eleq2i 2691 . . . . 5 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
38 ndmfv 6205 . . . . 5 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
3937, 38sylnbir 321 . . . 4 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
4039fveq2d 6182 . . 3 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
4133, 40, 393eqtr4a 2680 . 2 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
4232, 41pm2.61i 176 1 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  c0 3907   ciun 4511  dom cdm 5104  Oncon0 5711  Lim wlim 5712  suc csuc 5713   Fn wfn 5871  cfv 5876  ωcom 7050  harchar 8446  cardccrd 8746  cale 8747
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-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
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-reu 2916  df-rmo 2917  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-iun 4513  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-se 5064  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-pred 5668  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-isom 5885  df-riota 6596  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-har 8448  df-card 8750  df-aleph 8751
This theorem is referenced by:  alephnbtwn2  8880  alephord2  8884  alephsuc2  8888  alephislim  8891  alephsdom  8894  cardaleph  8897  cardalephex  8898  alephval3  8918  alephval2  9379  alephsuc3  9387  alephreg  9389  pwcfsdom  9390
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