Theorem alephon 8892

Description: An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Assertion

Ref	Expression
alephon	⊢ (ℵ‘𝐴) ∈ On

Proof of Theorem alephon

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephfnon 8888	. . 3 ⊢ ℵ Fn On
2		fveq2 6191	. . . . . 6 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	2	eleq1d 2686	. . . . 5 ⊢ (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On))
4		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
5	4	eleq1d 2686	. . . . 5 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On))
6		fveq2 6191	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
7	6	eleq1d 2686	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On))
8		aleph0 8889	. . . . . 6 ⊢ (ℵ‘∅) = ω
9		omelon 8543	. . . . . 6 ⊢ ω ∈ On
10	8, 9	eqeltri 2697	. . . . 5 ⊢ (ℵ‘∅) ∈ On
11		alephsuc 8891	. . . . . . 7 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
12		harcl 8466	. . . . . . 7 ⊢ (har‘(ℵ‘𝑦)) ∈ On
13	11, 12	syl6eqel 2709	. . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) ∈ On)
14	13	a1d 25	. . . . 5 ⊢ (𝑦 ∈ On → ((ℵ‘𝑦) ∈ On → (ℵ‘suc 𝑦) ∈ On))
15		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
16		iunon 7436	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On)
17	15, 16	mpan 706	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On)
18		alephlim 8890	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
19	15, 18	mpan 706	. . . . . . 7 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦))
20	19	eleq1d 2686	. . . . . 6 ⊢ (Lim 𝑥 → ((ℵ‘𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On))
21	17, 20	syl5ibr 236	. . . . 5 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → (ℵ‘𝑥) ∈ On))
22	3, 5, 7, 5, 10, 14, 21	tfinds 7059	. . . 4 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ On)
23	22	rgen 2922	. . 3 ⊢ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On
24		ffnfv 6388	. . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On))
25	1, 23, 24	mpbir2an 955	. 2 ⊢ ℵ:On⟶On
26		0elon 5778	. 2 ⊢ ∅ ∈ On
27	25, 26	f0cli 6370	1 ⊢ (ℵ‘𝐴) ∈ On

Syntax hints:   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  ∅c0 3915  ∪ ciun 4520  Oncon0 5723  Lim wlim 5724  suc csuc 5725   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  ωcom 7065  harchar 8461  ℵcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-en 7956  df-dom 7957  df-oi 8415  df-har 8463  df-aleph 8766

This theorem is referenced by:  alephnbtwn  8894  alephnbtwn2  8895  alephordilem1  8896  alephord  8898  alephord2  8899  alephord3  8901  alephsucdom  8902  alephsuc2  8903  alephf1  8908  alephsdom  8909  alephdom2  8910  alephle  8911  cardaleph  8912  alephf1ALT  8926  alephfp  8931  dfac12k  8969  alephsing  9098  alephval2  9394  alephadd  9399  alephmul  9400  alephexp1  9401  alephsuc3  9402  alephreg  9404  pwcfsdom  9405  cfpwsdom  9406  gchaleph  9493  gchaleph2  9494  gch2  9497