Theorem alephcard 9971

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.

Step	Hyp	Ref	Expression
1		2fveq3 6836	. . . 4 ⊢ (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
2		fveq2 6831	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	1, 2	eqeq12d 2749	. . 3 ⊢ (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4		2fveq3 6836	. . . 4 ⊢ (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
5		fveq2 6831	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
6	4, 5	eqeq12d 2749	. . 3 ⊢ (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7		2fveq3 6836	. . . 4 ⊢ (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
8		fveq2 6831	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
9	7, 8	eqeq12d 2749	. . 3 ⊢ (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10		2fveq3 6836	. . . 4 ⊢ (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
11		fveq2 6831	. . . 4 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
12	10, 11	eqeq12d 2749	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13		cardom 9889	. . . 4 ⊢ (card‘ω) = ω
14		aleph0 9967	. . . . 5 ⊢ (ℵ‘∅) = ω
15	14	fveq2i 6834	. . . 4 ⊢ (card‘(ℵ‘∅)) = (card‘ω)
16	13, 15, 14	3eqtr4i 2766	. . 3 ⊢ (card‘(ℵ‘∅)) = (ℵ‘∅)
17		harcard 9881	. . . . 5 ⊢ (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18		alephsuc 9969	. . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
19	18	fveq2d 6835	. . . . 5 ⊢ (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
20	17, 19, 18	3eqtr4a 2794	. . . 4 ⊢ (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦))
21	20	a1d 25	. . 3 ⊢ (𝑦 ∈ On → ((card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
22		cardiun 9885	. . . . . . 7 ⊢ (𝑥 ∈ V → (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
23	22	elv 3443	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
24	23	adantl 481	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
25		vex 3442	. . . . . . . 8 ⊢ 𝑥 ∈ V
26		alephlim 9968	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
27	25, 26	mpan 690	. . . . . . 7 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
28	27	adantr 480	. . . . . 6 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
29	28	fveq2d 6835	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
30	24, 29, 28	3eqtr4d 2778	. . . 4 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥))
31	30	ex 412	. . 3 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)))
32	3, 6, 9, 12, 16, 21, 31	tfinds 7799	. 2 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33		card0 9861	. . 3 ⊢ (card‘∅) = ∅
34		alephfnon 9966	. . . . . . 7 ⊢ ℵ Fn On
35	34	fndmi 6593	. . . . . 6 ⊢ dom ℵ = On
36	35	eleq2i 2825	. . . . 5 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
37		ndmfv 6863	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
38	36, 37	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
39	38	fveq2d 6835	. . 3 ⊢ (¬ 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
40	33, 39, 38	3eqtr4a 2794	. 2 ⊢ (¬ 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
41	32, 40	pm2.61i 182	1 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  Vcvv 3438  ∅c0 4284  ∪ ciun 4943  dom cdm 5621  Oncon0 6314  Lim wlim 6315  suc csuc 6316  ‘cfv 6489  ωcom 7805  harchar 9452  cardccrd 9838  ℵcale 9839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9541

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-oi 9406  df-har 9453  df-card 9842  df-aleph 9843

This theorem is referenced by:  alephnbtwn2  9973  alephord2  9977  alephsuc2  9981  alephislim  9984  alephsdom  9987  cardaleph  9990  cardalephex  9991  alephval3  10011  alephval2  10473  alephsuc3  10481  alephreg  10483  pwcfsdom  10484  minregex2  43642