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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 10102 | . . 3 ⊢ ℵ Fn On | |
2 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅)) | |
3 | 2 | eleq1d 2823 | . . . . 5 ⊢ (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On)) |
4 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦)) | |
5 | 4 | eleq1d 2823 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On)) |
6 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦)) | |
7 | 6 | eleq1d 2823 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On)) |
8 | aleph0 10103 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
9 | omelon 9683 | . . . . . 6 ⊢ ω ∈ On | |
10 | 8, 9 | eqeltri 2834 | . . . . 5 ⊢ (ℵ‘∅) ∈ On |
11 | alephsuc 10105 | . . . . . . 7 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦))) | |
12 | harcl 9596 | . . . . . . 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 3481 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
16 | iunon 8377 | . . . . . . 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 10104 | . . . . . . . 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 2823 | . . . . . 6 ⊢ (Lim 𝑥 → ((ℵ‘𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On)) |
21 | 17, 20 | imbitrrid 246 | . . . . 5 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → (ℵ‘𝑥) ∈ On)) |
22 | 3, 5, 7, 5, 10, 14, 21 | tfinds 7880 | . . . 4 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ On) |
23 | 22 | rgen 3060 | . . 3 ⊢ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On |
24 | ffnfv 7138 | . . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On)) | |
25 | 1, 23, 24 | mpbir2an 711 | . 2 ⊢ ℵ:On⟶On |
26 | 0elon 6439 | . 2 ⊢ ∅ ∈ On | |
27 | 25, 26 | f0cli 7117 | 1 ⊢ (ℵ‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 ∅c0 4338 ∪ ciun 4995 Oncon0 6385 Lim wlim 6386 suc csuc 6387 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 ωcom 7886 harchar 9593 ℵcale 9973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-en 8984 df-dom 8985 df-oi 9547 df-har 9594 df-aleph 9977 |
This theorem is referenced by: alephnbtwn 10108 alephnbtwn2 10109 alephordilem1 10110 alephord 10112 alephord2 10113 alephord3 10115 alephsucdom 10116 alephsuc2 10117 alephf1 10122 alephsdom 10123 alephdom2 10124 alephle 10125 cardaleph 10126 alephf1ALT 10140 alephfp 10145 dfac12k 10185 alephsing 10313 alephval2 10609 alephadd 10614 alephmul 10615 alephexp1 10616 alephsuc3 10617 alephreg 10619 pwcfsdom 10620 cfpwsdom 10621 gchaleph 10708 gchaleph2 10709 gch2 10712 minregex2 43524 alephiso2 43547 |
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