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Theorem alephcard 9093
 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 6332 . . . . 5 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
21fveq2d 6336 . . . 4 (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
32, 1eqeq12d 2786 . . 3 (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4 fveq2 6332 . . . . 5 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
54fveq2d 6336 . . . 4 (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
65, 4eqeq12d 2786 . . 3 (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7 fveq2 6332 . . . . 5 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
87fveq2d 6336 . . . 4 (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
98, 7eqeq12d 2786 . . 3 (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10 fveq2 6332 . . . . 5 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
1110fveq2d 6336 . . . 4 (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
1211, 10eqeq12d 2786 . . 3 (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13 cardom 9012 . . . 4 (card‘ω) = ω
14 aleph0 9089 . . . . 5 (ℵ‘∅) = ω
1514fveq2i 6335 . . . 4 (card‘(ℵ‘∅)) = (card‘ω)
1613, 15, 143eqtr4i 2803 . . 3 (card‘(ℵ‘∅)) = (ℵ‘∅)
17 harcard 9004 . . . . 5 (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18 alephsuc 9091 . . . . . 6 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
1918fveq2d 6336 . . . . 5 (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
2017, 19, 183eqtr4a 2831 . . . 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 3354 . . . . . . 7 𝑥 ∈ V
23 cardiun 9008 . . . . . . 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 467 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘ 𝑦𝑥 (ℵ‘𝑦)) = 𝑦𝑥 (ℵ‘𝑦))
26 alephlim 9090 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2722, 26mpan 670 . . . . . . 7 (Lim 𝑥 → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2827adantr 466 . . . . . 6 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2928fveq2d 6336 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘ 𝑦𝑥 (ℵ‘𝑦)))
3025, 29, 283eqtr4d 2815 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥))
3130ex 397 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)))
323, 6, 9, 12, 16, 21, 31tfinds 7206 . 2 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33 card0 8984 . . 3 (card‘∅) = ∅
34 alephfnon 9088 . . . . . . 7 ℵ Fn On
35 fndm 6130 . . . . . . 7 (ℵ Fn On → dom ℵ = On)
3634, 35ax-mp 5 . . . . . 6 dom ℵ = On
3736eleq2i 2842 . . . . 5 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
38 ndmfv 6359 . . . . 5 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
3937, 38sylnbir 320 . . . 4 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
4039fveq2d 6336 . . 3 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
4133, 40, 393eqtr4a 2831 . 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 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351  ∅c0 4063  ∪ ciun 4654  dom cdm 5249  Oncon0 5866  Lim wlim 5867  suc csuc 5868   Fn wfn 6026  ‘cfv 6031  ωcom 7212  harchar 8617  cardccrd 8961  ℵcale 8962 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-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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-reu 3068  df-rmo 3069  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-iun 4656  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-se 5209  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-pred 5823  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-isom 6040  df-riota 6754  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-oi 8571  df-har 8619  df-card 8965  df-aleph 8966 This theorem is referenced by:  alephnbtwn2  9095  alephord2  9099  alephsuc2  9103  alephislim  9106  alephsdom  9109  cardaleph  9112  cardalephex  9113  alephval3  9133  alephval2  9596  alephsuc3  9604  alephreg  9606  pwcfsdom  9607
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