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Theorem alephcard 8754
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 6088 . . . . 5 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
21fveq2d 6092 . . . 4 (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
32, 1eqeq12d 2624 . . 3 (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4 fveq2 6088 . . . . 5 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
54fveq2d 6092 . . . 4 (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
65, 4eqeq12d 2624 . . 3 (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7 fveq2 6088 . . . . 5 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
87fveq2d 6092 . . . 4 (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
98, 7eqeq12d 2624 . . 3 (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10 fveq2 6088 . . . . 5 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
1110fveq2d 6092 . . . 4 (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
1211, 10eqeq12d 2624 . . 3 (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13 cardom 8673 . . . 4 (card‘ω) = ω
14 aleph0 8750 . . . . 5 (ℵ‘∅) = ω
1514fveq2i 6091 . . . 4 (card‘(ℵ‘∅)) = (card‘ω)
1613, 15, 143eqtr4i 2641 . . 3 (card‘(ℵ‘∅)) = (ℵ‘∅)
17 harcard 8665 . . . . 5 (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18 alephsuc 8752 . . . . . 6 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
1918fveq2d 6092 . . . . 5 (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
2017, 19, 183eqtr4a 2669 . . . 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 3175 . . . . . . 7 𝑥 ∈ V
23 cardiun 8669 . . . . . . 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 480 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘ 𝑦𝑥 (ℵ‘𝑦)) = 𝑦𝑥 (ℵ‘𝑦))
26 alephlim 8751 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2722, 26mpan 701 . . . . . . 7 (Lim 𝑥 → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2827adantr 479 . . . . . 6 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2928fveq2d 6092 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘ 𝑦𝑥 (ℵ‘𝑦)))
3025, 29, 283eqtr4d 2653 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥))
3130ex 448 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)))
323, 6, 9, 12, 16, 21, 31tfinds 6929 . 2 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33 card0 8645 . . 3 (card‘∅) = ∅
34 alephfnon 8749 . . . . . . 7 ℵ Fn On
35 fndm 5890 . . . . . . 7 (ℵ Fn On → dom ℵ = On)
3634, 35ax-mp 5 . . . . . 6 dom ℵ = On
3736eleq2i 2679 . . . . 5 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
38 ndmfv 6113 . . . . 5 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
3937, 38sylnbir 319 . . . 4 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
4039fveq2d 6092 . . 3 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
4133, 40, 393eqtr4a 2669 . 2 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
4232, 41pm2.61i 174 1 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  c0 3873   ciun 4449  dom cdm 5028  Oncon0 5626  Lim wlim 5627  suc csuc 5628   Fn wfn 5785  cfv 5790  ωcom 6935  harchar 8322  cardccrd 8622  cale 8623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-om 6936  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-oi 8276  df-har 8324  df-card 8626  df-aleph 8627
This theorem is referenced by:  alephnbtwn2  8756  alephord2  8760  alephsuc2  8764  alephislim  8767  alephsdom  8770  cardaleph  8773  cardalephex  8774  alephval3  8794  alephval2  9251  alephsuc3  9259  alephreg  9261  pwcfsdom  9262
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