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Theorem alephcard 8776
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 6103 . . . . 5 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
21fveq2d 6107 . . . 4 (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
32, 1eqeq12d 2625 . . 3 (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4 fveq2 6103 . . . . 5 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
54fveq2d 6107 . . . 4 (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
65, 4eqeq12d 2625 . . 3 (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7 fveq2 6103 . . . . 5 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
87fveq2d 6107 . . . 4 (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
98, 7eqeq12d 2625 . . 3 (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10 fveq2 6103 . . . . 5 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
1110fveq2d 6107 . . . 4 (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
1211, 10eqeq12d 2625 . . 3 (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13 cardom 8695 . . . 4 (card‘ω) = ω
14 aleph0 8772 . . . . 5 (ℵ‘∅) = ω
1514fveq2i 6106 . . . 4 (card‘(ℵ‘∅)) = (card‘ω)
1613, 15, 143eqtr4i 2642 . . 3 (card‘(ℵ‘∅)) = (ℵ‘∅)
17 harcard 8687 . . . . 5 (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18 alephsuc 8774 . . . . . 6 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
1918fveq2d 6107 . . . . 5 (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
2017, 19, 183eqtr4a 2670 . . . 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 3176 . . . . . . 7 𝑥 ∈ V
23 cardiun 8691 . . . . . . 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 481 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘ 𝑦𝑥 (ℵ‘𝑦)) = 𝑦𝑥 (ℵ‘𝑦))
26 alephlim 8773 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2722, 26mpan 702 . . . . . . 7 (Lim 𝑥 → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2827adantr 480 . . . . . 6 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2928fveq2d 6107 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘ 𝑦𝑥 (ℵ‘𝑦)))
3025, 29, 283eqtr4d 2654 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥))
3130ex 449 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)))
323, 6, 9, 12, 16, 21, 31tfinds 6951 . 2 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33 card0 8667 . . 3 (card‘∅) = ∅
34 alephfnon 8771 . . . . . . 7 ℵ Fn On
35 fndm 5904 . . . . . . 7 (ℵ Fn On → dom ℵ = On)
3634, 35ax-mp 5 . . . . . 6 dom ℵ = On
3736eleq2i 2680 . . . . 5 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
38 ndmfv 6128 . . . . 5 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
3937, 38sylnbir 320 . . . 4 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
4039fveq2d 6107 . . 3 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
4133, 40, 393eqtr4a 2670 . 2 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
4232, 41pm2.61i 175 1 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  c0 3874   ciun 4455  dom cdm 5038  Oncon0 5640  Lim wlim 5641  suc csuc 5642   Fn wfn 5799  cfv 5804  ωcom 6957  harchar 8344  cardccrd 8644  cale 8645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649
This theorem is referenced by:  alephnbtwn2  8778  alephord2  8782  alephsuc2  8786  alephislim  8789  alephsdom  8792  cardaleph  8795  cardalephex  8796  alephval3  8816  alephval2  9273  alephsuc3  9281  alephreg  9283  pwcfsdom  9284
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