Theorem alephcard 10150

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 6890	. . . 4 ⊢ (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
2		fveq2 6885	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	1, 2	eqeq12d 2776	. . 3 ⊢ (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4		2fveq3 6890	. . . 4 ⊢ (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
5		fveq2 6885	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
6	4, 5	eqeq12d 2776	. . 3 ⊢ (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7		2fveq3 6890	. . . 4 ⊢ (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
8		fveq2 6885	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
9	7, 8	eqeq12d 2776	. . 3 ⊢ (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10		2fveq3 6890	. . . 4 ⊢ (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
11		fveq2 6885	. . . 4 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
12	10, 11	eqeq12d 2776	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13		cardom 10068	. . . 4 ⊢ (card‘ω) = ω
14		aleph0 10146	. . . . 5 ⊢ (ℵ‘∅) = ω
15	14	fveq2i 6888	. . . 4 ⊢ (card‘(ℵ‘∅)) = (card‘ω)
16	13, 15, 14	3eqtr4i 2793	. . 3 ⊢ (card‘(ℵ‘∅)) = (ℵ‘∅)
17		harcard 10060	. . . . 5 ⊢ (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18		alephsuc 10148	. . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
19	18	fveq2d 6889	. . . . 5 ⊢ (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
20	17, 19, 18	3eqtr4a 2821	. . . 4 ⊢ (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦))
21	20	a1d 26	. . 3 ⊢ (𝑦 ∈ On → ((card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
22		cardiun 10064	. . . . . . 7 ⊢ (𝑥 ∈ V → (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
23	22	elv 3455	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
24	23	adantl 487	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
25		vex 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
26		alephlim 10147	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
27	25, 26	mpan 703	. . . . . . 7 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
28	27	adantr 486	. . . . . 6 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
29	28	fveq2d 6889	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
30	24, 29, 28	3eqtr4d 2805	. . . 4 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥))
31	30	ex 418	. . 3 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)))
32	3, 6, 9, 12, 16, 21, 31	tfinds 7871	. 2 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33		card0 10040	. . 3 ⊢ (card‘∅) = ∅
34		alephfnon 10145	. . . . . . 7 ⊢ ℵ Fn On
35	34	fndmi 6643	. . . . . 6 ⊢ dom ℵ = On
36	35	eleq2i 2852	. . . . 5 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
37		ndmfv 6917	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
38	36, 37	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
39	38	fveq2d 6889	. . 3 ⊢ (¬ 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
40	33, 39, 38	3eqtr4a 2821	. 2 ⊢ (¬ 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
41	32, 40	pm2.61i 184	1 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450  ∅c0 4278  ∪ ciun 4950  dom cdm 5650  Oncon0 6362  Lim wlim 6363  suc csuc 6364  ‘cfv 6538  ωcom 7877  harchar 9551  cardccrd 10017  ℵcale 10018

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9643

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-er 8718  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-oi 9505  df-har 9552  df-card 10021  df-aleph 10022

This theorem is used by:  alephnbtwn2  10152  alephord2  10156  alephsuc2  10160  alephislim  10163  alephsdom  10166  cardaleph  10169  cardalephex  10170  alephval3  10190  alephval2  10658  alephsuc3  10666  alephreg  10668  pwcfsdom  10669  minregex2  44612