Theorem alephcard 9498

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 6677	. . . 4 ⊢ (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
2		fveq2 6672	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	1, 2	eqeq12d 2839	. . 3 ⊢ (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4		2fveq3 6677	. . . 4 ⊢ (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
5		fveq2 6672	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
6	4, 5	eqeq12d 2839	. . 3 ⊢ (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7		2fveq3 6677	. . . 4 ⊢ (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
8		fveq2 6672	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
9	7, 8	eqeq12d 2839	. . 3 ⊢ (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10		2fveq3 6677	. . . 4 ⊢ (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
11		fveq2 6672	. . . 4 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
12	10, 11	eqeq12d 2839	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13		cardom 9417	. . . 4 ⊢ (card‘ω) = ω
14		aleph0 9494	. . . . 5 ⊢ (ℵ‘∅) = ω
15	14	fveq2i 6675	. . . 4 ⊢ (card‘(ℵ‘∅)) = (card‘ω)
16	13, 15, 14	3eqtr4i 2856	. . 3 ⊢ (card‘(ℵ‘∅)) = (ℵ‘∅)
17		harcard 9409	. . . . 5 ⊢ (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18		alephsuc 9496	. . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
19	18	fveq2d 6676	. . . . 5 ⊢ (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
20	17, 19, 18	3eqtr4a 2884	. . . 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 9413	. . . . . . 7 ⊢ (𝑥 ∈ V → (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
23	22	elv 3501	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
24	23	adantl 484	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
25		vex 3499	. . . . . . . 8 ⊢ 𝑥 ∈ V
26		alephlim 9495	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
27	25, 26	mpan 688	. . . . . . 7 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
28	27	adantr 483	. . . . . 6 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
29	28	fveq2d 6676	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
30	24, 29, 28	3eqtr4d 2868	. . . 4 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥))
31	30	ex 415	. . 3 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)))
32	3, 6, 9, 12, 16, 21, 31	tfinds 7576	. 2 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33		card0 9389	. . 3 ⊢ (card‘∅) = ∅
34		alephfnon 9493	. . . . . . 7 ⊢ ℵ Fn On
35		fndm 6457	. . . . . . 7 ⊢ (ℵ Fn On → dom ℵ = On)
36	34, 35	ax-mp 5	. . . . . 6 ⊢ dom ℵ = On
37	36	eleq2i 2906	. . . . 5 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
38		ndmfv 6702	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
39	37, 38	sylnbir 333	. . . 4 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
40	39	fveq2d 6676	. . 3 ⊢ (¬ 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
41	33, 40, 39	3eqtr4a 2884	. 2 ⊢ (¬ 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
42	32, 41	pm2.61i 184	1 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496  ∅c0 4293  ∪ ciun 4921  dom cdm 5557  Oncon0 6193  Lim wlim 6194  suc csuc 6195   Fn wfn 6352  ‘cfv 6357  ωcom 7582  harchar 9022  cardccrd 9366  ℵcale 9367

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-oi 8976  df-har 9024  df-card 9370  df-aleph 9371

This theorem is referenced by:  alephnbtwn2  9500  alephord2  9504  alephsuc2  9508  alephislim  9511  alephsdom  9514  cardaleph  9517  cardalephex  9518  alephval3  9538  alephval2  9996  alephsuc3  10004  alephreg  10006  pwcfsdom  10007