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Theorem alephon 9113
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.
StepHypRef Expression
1 alephfnon 9109 . . 3 ℵ Fn On
2 fveq2 6348 . . . . . 6 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
32eleq1d 2838 . . . . 5 (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On))
4 fveq2 6348 . . . . . 6 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
54eleq1d 2838 . . . . 5 (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On))
6 fveq2 6348 . . . . . 6 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
76eleq1d 2838 . . . . 5 (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On))
8 aleph0 9110 . . . . . 6 (ℵ‘∅) = ω
9 omelon 8728 . . . . . 6 ω ∈ On
108, 9eqeltri 2849 . . . . 5 (ℵ‘∅) ∈ On
11 alephsuc 9112 . . . . . . 7 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
12 harcl 8643 . . . . . . 7 (har‘(ℵ‘𝑦)) ∈ On
1311, 12syl6eqel 2861 . . . . . 6 (𝑦 ∈ On → (ℵ‘suc 𝑦) ∈ On)
1413a1d 25 . . . . 5 (𝑦 ∈ On → ((ℵ‘𝑦) ∈ On → (ℵ‘suc 𝑦) ∈ On))
15 vex 3358 . . . . . . 7 𝑥 ∈ V
16 iunon 7610 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (ℵ‘𝑦) ∈ On) → 𝑦𝑥 (ℵ‘𝑦) ∈ On)
1715, 16mpan 671 . . . . . 6 (∀𝑦𝑥 (ℵ‘𝑦) ∈ On → 𝑦𝑥 (ℵ‘𝑦) ∈ On)
18 alephlim 9111 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
1915, 18mpan 671 . . . . . . 7 (Lim 𝑥 → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2019eleq1d 2838 . . . . . 6 (Lim 𝑥 → ((ℵ‘𝑥) ∈ On ↔ 𝑦𝑥 (ℵ‘𝑦) ∈ On))
2117, 20syl5ibr 237 . . . . 5 (Lim 𝑥 → (∀𝑦𝑥 (ℵ‘𝑦) ∈ On → (ℵ‘𝑥) ∈ On))
223, 5, 7, 5, 10, 14, 21tfinds 7227 . . . 4 (𝑦 ∈ On → (ℵ‘𝑦) ∈ On)
2322rgen 3074 . . 3 𝑦 ∈ On (ℵ‘𝑦) ∈ On
24 ffnfv 6548 . . 3 (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On))
251, 23, 24mpbir2an 691 . 2 ℵ:On⟶On
26 0elon 5932 . 2 ∅ ∈ On
2725, 26f0cli 6530 1 (ℵ‘𝐴) ∈ On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1634  wcel 2148  wral 3064  Vcvv 3355  c0 4073   ciun 4665  Oncon0 5877  Lim wlim 5878  suc csuc 5879   Fn wfn 6037  wf 6038  cfv 6042  ωcom 7233  harchar 8638  cale 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-inf2 8723
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-se 5223  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-isom 6051  df-riota 6773  df-om 7234  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-en 8131  df-dom 8132  df-oi 8592  df-har 8640  df-aleph 8987
This theorem is referenced by:  alephnbtwn  9115  alephnbtwn2  9116  alephordilem1  9117  alephord  9119  alephord2  9120  alephord3  9122  alephsucdom  9123  alephsuc2  9124  alephf1  9129  alephsdom  9130  alephdom2  9131  alephle  9132  cardaleph  9133  alephf1ALT  9147  alephfp  9152  dfac12k  9192  alephsing  9321  alephval2  9617  alephadd  9622  alephmul  9623  alephexp1  9624  alephsuc3  9625  alephreg  9627  pwcfsdom  9628  cfpwsdom  9629  gchaleph  9716  gchaleph2  9717  gch2  9720
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