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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 9684 | . . 3 ⊢ ℵ Fn On | |
2 | fveq2 6722 | . . . . . 6 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅)) | |
3 | 2 | eleq1d 2822 | . . . . 5 ⊢ (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On)) |
4 | fveq2 6722 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦)) | |
5 | 4 | eleq1d 2822 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On)) |
6 | fveq2 6722 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦)) | |
7 | 6 | eleq1d 2822 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On)) |
8 | aleph0 9685 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
9 | omelon 9266 | . . . . . 6 ⊢ ω ∈ On | |
10 | 8, 9 | eqeltri 2834 | . . . . 5 ⊢ (ℵ‘∅) ∈ On |
11 | alephsuc 9687 | . . . . . . 7 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦))) | |
12 | harcl 9180 | . . . . . . 7 ⊢ (har‘(ℵ‘𝑦)) ∈ On | |
13 | 11, 12 | eqeltrdi 2846 | . . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) ∈ On) |
14 | 13 | a1d 25 | . . . . 5 ⊢ (𝑦 ∈ On → ((ℵ‘𝑦) ∈ On → (ℵ‘suc 𝑦) ∈ On)) |
15 | vex 3417 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
16 | iunon 8081 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) | |
17 | 15, 16 | mpan 690 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) |
18 | alephlim 9686 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) | |
19 | 15, 18 | mpan 690 | . . . . . . 7 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) |
20 | 19 | eleq1d 2822 | . . . . . 6 ⊢ (Lim 𝑥 → ((ℵ‘𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On)) |
21 | 17, 20 | syl5ibr 249 | . . . . 5 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → (ℵ‘𝑥) ∈ On)) |
22 | 3, 5, 7, 5, 10, 14, 21 | tfinds 7643 | . . . 4 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ On) |
23 | 22 | rgen 3071 | . . 3 ⊢ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On |
24 | ffnfv 6940 | . . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On)) | |
25 | 1, 23, 24 | mpbir2an 711 | . 2 ⊢ ℵ:On⟶On |
26 | 0elon 6271 | . 2 ⊢ ∅ ∈ On | |
27 | 25, 26 | f0cli 6922 | 1 ⊢ (ℵ‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3413 ∅c0 4242 ∪ ciun 4909 Oncon0 6218 Lim wlim 6219 suc csuc 6220 Fn wfn 6380 ⟶wf 6381 ‘cfv 6385 ωcom 7649 harchar 9177 ℵcale 9557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-inf2 9261 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-se 5515 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-isom 6394 df-riota 7175 df-om 7650 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-en 8632 df-dom 8633 df-oi 9131 df-har 9178 df-aleph 9561 |
This theorem is referenced by: alephnbtwn 9690 alephnbtwn2 9691 alephordilem1 9692 alephord 9694 alephord2 9695 alephord3 9697 alephsucdom 9698 alephsuc2 9699 alephf1 9704 alephsdom 9705 alephdom2 9706 alephle 9707 cardaleph 9708 alephf1ALT 9722 alephfp 9727 dfac12k 9766 alephsing 9895 alephval2 10191 alephadd 10196 alephmul 10197 alephexp1 10198 alephsuc3 10199 alephreg 10201 pwcfsdom 10202 cfpwsdom 10203 gchaleph 10290 gchaleph2 10291 gch2 10294 alephiso2 40849 |
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