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Theorem alephcard 9483
 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 2fveq3 6650 . . . 4 (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
2 fveq2 6645 . . . 4 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
31, 2eqeq12d 2814 . . 3 (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4 2fveq3 6650 . . . 4 (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
5 fveq2 6645 . . . 4 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
64, 5eqeq12d 2814 . . 3 (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7 2fveq3 6650 . . . 4 (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
8 fveq2 6645 . . . 4 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
97, 8eqeq12d 2814 . . 3 (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10 2fveq3 6650 . . . 4 (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
11 fveq2 6645 . . . 4 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
1210, 11eqeq12d 2814 . . 3 (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13 cardom 9401 . . . 4 (card‘ω) = ω
14 aleph0 9479 . . . . 5 (ℵ‘∅) = ω
1514fveq2i 6648 . . . 4 (card‘(ℵ‘∅)) = (card‘ω)
1613, 15, 143eqtr4i 2831 . . 3 (card‘(ℵ‘∅)) = (ℵ‘∅)
17 harcard 9393 . . . . 5 (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18 alephsuc 9481 . . . . . 6 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
1918fveq2d 6649 . . . . 5 (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
2017, 19, 183eqtr4a 2859 . . . 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 cardiun 9397 . . . . . . 7 (𝑥 ∈ V → (∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘ 𝑦𝑥 (ℵ‘𝑦)) = 𝑦𝑥 (ℵ‘𝑦)))
2322elv 3446 . . . . . 6 (∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘ 𝑦𝑥 (ℵ‘𝑦)) = 𝑦𝑥 (ℵ‘𝑦))
2423adantl 485 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘ 𝑦𝑥 (ℵ‘𝑦)) = 𝑦𝑥 (ℵ‘𝑦))
25 vex 3444 . . . . . . . 8 𝑥 ∈ V
26 alephlim 9480 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2725, 26mpan 689 . . . . . . 7 (Lim 𝑥 → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2827adantr 484 . . . . . 6 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2928fveq2d 6649 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘ 𝑦𝑥 (ℵ‘𝑦)))
3024, 29, 283eqtr4d 2843 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥))
3130ex 416 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)))
323, 6, 9, 12, 16, 21, 31tfinds 7556 . 2 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33 card0 9373 . . 3 (card‘∅) = ∅
34 alephfnon 9478 . . . . . . 7 ℵ Fn On
3534fndmi 6426 . . . . . 6 dom ℵ = On
3635eleq2i 2881 . . . . 5 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
37 ndmfv 6675 . . . . 5 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
3836, 37sylnbir 334 . . . 4 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
3938fveq2d 6649 . . 3 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
4033, 39, 383eqtr4a 2859 . 2 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
4132, 40pm2.61i 185 1 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ∅c0 4243  ∪ ciun 4881  dom cdm 5519  Oncon0 6159  Lim wlim 6160  suc csuc 6161  ‘cfv 6324  ωcom 7562  harchar 9006  cardccrd 9350  ℵcale 9351 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 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-inf2 9090 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-om 7563  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-oi 8960  df-har 9007  df-card 9354  df-aleph 9355 This theorem is referenced by:  alephnbtwn2  9485  alephord2  9489  alephsuc2  9493  alephislim  9496  alephsdom  9499  cardaleph  9502  cardalephex  9503  alephval3  9523  alephval2  9985  alephsuc3  9993  alephreg  9995  pwcfsdom  9996
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