Theorem alephcard 9342

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 6543	. . . 4 ⊢ (𝑥 = ∅ → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘∅)))
2		fveq2 6538	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	1, 2	eqeq12d 2810	. . 3 ⊢ (𝑥 = ∅ → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘∅)) = (ℵ‘∅)))
4		2fveq3 6543	. . . 4 ⊢ (𝑥 = 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝑦)))
5		fveq2 6538	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
6	4, 5	eqeq12d 2810	. . 3 ⊢ (𝑥 = 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
7		2fveq3 6543	. . . 4 ⊢ (𝑥 = suc 𝑦 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘suc 𝑦)))
8		fveq2 6538	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
9	7, 8	eqeq12d 2810	. . 3 ⊢ (𝑥 = suc 𝑦 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘suc 𝑦)) = (ℵ‘suc 𝑦)))
10		2fveq3 6543	. . . 4 ⊢ (𝑥 = 𝐴 → (card‘(ℵ‘𝑥)) = (card‘(ℵ‘𝐴)))
11		fveq2 6538	. . . 4 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
12	10, 11	eqeq12d 2810	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘(ℵ‘𝑥)) = (ℵ‘𝑥) ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
13		cardom 9261	. . . 4 ⊢ (card‘ω) = ω
14		aleph0 9338	. . . . 5 ⊢ (ℵ‘∅) = ω
15	14	fveq2i 6541	. . . 4 ⊢ (card‘(ℵ‘∅)) = (card‘ω)
16	13, 15, 14	3eqtr4i 2829	. . 3 ⊢ (card‘(ℵ‘∅)) = (ℵ‘∅)
17		harcard 9253	. . . . 5 ⊢ (card‘(har‘(ℵ‘𝑦))) = (har‘(ℵ‘𝑦))
18		alephsuc 9340	. . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
19	18	fveq2d 6542	. . . . 5 ⊢ (𝑦 ∈ On → (card‘(ℵ‘suc 𝑦)) = (card‘(har‘(ℵ‘𝑦))))
20	17, 19, 18	3eqtr4a 2857	. . . 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 9257	. . . . . . 7 ⊢ (𝑥 ∈ V → (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
23	22	elv 3442	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
24	23	adantl 482	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
25		vex 3440	. . . . . . . 8 ⊢ 𝑥 ∈ V
26		alephlim 9339	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
27	25, 26	mpan 686	. . . . . . 7 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
28	27	adantr 481	. . . . . 6 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
29	28	fveq2d 6542	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (card‘∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)))
30	24, 29, 28	3eqtr4d 2841	. . . 4 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥))
31	30	ex 413	. . 3 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦) → (card‘(ℵ‘𝑥)) = (ℵ‘𝑥)))
32	3, 6, 9, 12, 16, 21, 31	tfinds 7430	. 2 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
33		card0 9233	. . 3 ⊢ (card‘∅) = ∅
34		alephfnon 9337	. . . . . . 7 ⊢ ℵ Fn On
35		fndm 6325	. . . . . . 7 ⊢ (ℵ Fn On → dom ℵ = On)
36	34, 35	ax-mp 5	. . . . . 6 ⊢ dom ℵ = On
37	36	eleq2i 2874	. . . . 5 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
38		ndmfv 6568	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
39	37, 38	sylnbir 332	. . . 4 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
40	39	fveq2d 6542	. . 3 ⊢ (¬ 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (card‘∅))
41	33, 40, 39	3eqtr4a 2857	. 2 ⊢ (¬ 𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
42	32, 41	pm2.61i 183	1 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105  Vcvv 3437  ∅c0 4211  ∪ ciun 4825  dom cdm 5443  Oncon0 6066  Lim wlim 6067  suc csuc 6068   Fn wfn 6220  ‘cfv 6225  ωcom 7436  harchar 8866  cardccrd 9210  ℵcale 9211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-oi 8820  df-har 8868  df-card 9214  df-aleph 9215

This theorem is referenced by:  alephnbtwn2  9344  alephord2  9348  alephsuc2  9352  alephislim  9355  alephsdom  9358  cardaleph  9361  cardalephex  9362  alephval3  9382  alephval2  9840  alephsuc3  9848  alephreg  9850  pwcfsdom  9851