Theorem alephcard 10069

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 6896	. . . 4 ⊢ (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
2		fveq2 6891	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	1, 2	eqeq12d 2747	. . 3 ⊢ (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4		2fveq3 6896	. . . 4 ⊢ (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
5		fveq2 6891	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
6	4, 5	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7		2fveq3 6896	. . . 4 ⊢ (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
8		fveq2 6891	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
9	7, 8	eqeq12d 2747	. . 3 ⊢ (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10		2fveq3 6896	. . . 4 ⊢ (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
11		fveq2 6891	. . . 4 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
12	10, 11	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13		cardom 9985	. . . 4 ⊢ (card‘ω) = ω
14		aleph0 10065	. . . . 5 ⊢ (ℵ‘∅) = ω
15	14	fveq2i 6894	. . . 4 ⊢ (card‘(ℵ‘∅)) = (card‘ω)
16	13, 15, 14	3eqtr4i 2769	. . 3 ⊢ (card‘(ℵ‘∅)) = (ℵ‘∅)
17		harcard 9977	. . . . 5 ⊢ (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18		alephsuc 10067	. . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
19	18	fveq2d 6895	. . . . 5 ⊢ (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
20	17, 19, 18	3eqtr4a 2797	. . . 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 9981	. . . . . . 7 ⊢ (𝑥 ∈ V → (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
23	22	elv 3479	. . . . . 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 3477	. . . . . . . 8 ⊢ 𝑥 ∈ V
26		alephlim 10066	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
27	25, 26	mpan 687	. . . . . . 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 6895	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
30	24, 29, 28	3eqtr4d 2781	. . . 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 7853	. 2 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33		card0 9957	. . 3 ⊢ (card‘∅) = ∅
34		alephfnon 10064	. . . . . . 7 ⊢ ℵ Fn On
35	34	fndmi 6653	. . . . . 6 ⊢ dom ℵ = On
36	35	eleq2i 2824	. . . . 5 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
37		ndmfv 6926	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
38	36, 37	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
39	38	fveq2d 6895	. . 3 ⊢ (¬ 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
40	33, 39, 38	3eqtr4a 2797	. 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 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473  ∅c0 4322  ∪ ciun 4997  dom cdm 5676  Oncon0 6364  Lim wlim 6365  suc csuc 6366  ‘cfv 6543  ωcom 7859  harchar 9555  cardccrd 9934  ℵcale 9935

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9640

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-oi 9509  df-har 9556  df-card 9938  df-aleph 9939

This theorem is referenced by:  alephnbtwn2  10071  alephord2  10075  alephsuc2  10079  alephislim  10082  alephsdom  10085  cardaleph  10088  cardalephex  10089  alephval3  10109  alephval2  10571  alephsuc3  10579  alephreg  10581  pwcfsdom  10582  minregex2  42589