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Mirrors > Home > MPE Home > Th. List > alephon | Structured version Visualization version GIF version |
Description: An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
alephon | ⊢ (ℵ‘𝐴) ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephfnon 9491 | . . 3 ⊢ ℵ Fn On | |
2 | fveq2 6670 | . . . . . 6 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅)) | |
3 | 2 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On)) |
4 | fveq2 6670 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦)) | |
5 | 4 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On)) |
6 | fveq2 6670 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦)) | |
7 | 6 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On)) |
8 | aleph0 9492 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
9 | omelon 9109 | . . . . . 6 ⊢ ω ∈ On | |
10 | 8, 9 | eqeltri 2909 | . . . . 5 ⊢ (ℵ‘∅) ∈ On |
11 | alephsuc 9494 | . . . . . . 7 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦))) | |
12 | harcl 9025 | . . . . . . 7 ⊢ (har‘(ℵ‘𝑦)) ∈ On | |
13 | 11, 12 | eqeltrdi 2921 | . . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) ∈ On) |
14 | 13 | a1d 25 | . . . . 5 ⊢ (𝑦 ∈ On → ((ℵ‘𝑦) ∈ On → (ℵ‘suc 𝑦) ∈ On)) |
15 | vex 3497 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
16 | iunon 7976 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) | |
17 | 15, 16 | mpan 688 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) |
18 | alephlim 9493 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) | |
19 | 15, 18 | mpan 688 | . . . . . . 7 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) |
20 | 19 | eleq1d 2897 | . . . . . 6 ⊢ (Lim 𝑥 → ((ℵ‘𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On)) |
21 | 17, 20 | syl5ibr 248 | . . . . 5 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → (ℵ‘𝑥) ∈ On)) |
22 | 3, 5, 7, 5, 10, 14, 21 | tfinds 7574 | . . . 4 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ On) |
23 | 22 | rgen 3148 | . . 3 ⊢ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On |
24 | ffnfv 6882 | . . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On)) | |
25 | 1, 23, 24 | mpbir2an 709 | . 2 ⊢ ℵ:On⟶On |
26 | 0elon 6244 | . 2 ⊢ ∅ ∈ On | |
27 | 25, 26 | f0cli 6864 | 1 ⊢ (ℵ‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ∅c0 4291 ∪ ciun 4919 Oncon0 6191 Lim wlim 6192 suc csuc 6193 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 ωcom 7580 harchar 9020 ℵcale 9365 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-en 8510 df-dom 8511 df-oi 8974 df-har 9022 df-aleph 9369 |
This theorem is referenced by: alephnbtwn 9497 alephnbtwn2 9498 alephordilem1 9499 alephord 9501 alephord2 9502 alephord3 9504 alephsucdom 9505 alephsuc2 9506 alephf1 9511 alephsdom 9512 alephdom2 9513 alephle 9514 cardaleph 9515 alephf1ALT 9529 alephfp 9534 dfac12k 9573 alephsing 9698 alephval2 9994 alephadd 9999 alephmul 10000 alephexp1 10001 alephsuc3 10002 alephreg 10004 pwcfsdom 10005 cfpwsdom 10006 gchaleph 10093 gchaleph2 10094 gch2 10097 alephiso2 39937 |
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