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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 10105 | . . 3 ⊢ ℵ Fn On | |
| 2 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅)) | |
| 3 | 2 | eleq1d 2826 | . . . . 5 ⊢ (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On)) |
| 4 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦)) | |
| 5 | 4 | eleq1d 2826 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On)) |
| 6 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦)) | |
| 7 | 6 | eleq1d 2826 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On)) |
| 8 | aleph0 10106 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
| 9 | omelon 9686 | . . . . . 6 ⊢ ω ∈ On | |
| 10 | 8, 9 | eqeltri 2837 | . . . . 5 ⊢ (ℵ‘∅) ∈ On |
| 11 | alephsuc 10108 | . . . . . . 7 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦))) | |
| 12 | harcl 9599 | . . . . . . 7 ⊢ (har‘(ℵ‘𝑦)) ∈ On | |
| 13 | 11, 12 | eqeltrdi 2849 | . . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) ∈ On) |
| 14 | 13 | a1d 25 | . . . . 5 ⊢ (𝑦 ∈ On → ((ℵ‘𝑦) ∈ On → (ℵ‘suc 𝑦) ∈ On)) |
| 15 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 16 | iunon 8379 | . . . . . . 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 10107 | . . . . . . . 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 2826 | . . . . . 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 7881 | . . . 4 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ On) |
| 23 | 22 | rgen 3063 | . . 3 ⊢ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On |
| 24 | ffnfv 7139 | . . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On)) | |
| 25 | 1, 23, 24 | mpbir2an 711 | . 2 ⊢ ℵ:On⟶On |
| 26 | 0elon 6438 | . 2 ⊢ ∅ ∈ On | |
| 27 | 25, 26 | f0cli 7118 | 1 ⊢ (ℵ‘𝐴) ∈ On |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∅c0 4333 ∪ ciun 4991 Oncon0 6384 Lim wlim 6385 suc csuc 6386 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 ωcom 7887 harchar 9596 ℵcale 9976 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-en 8986 df-dom 8987 df-oi 9550 df-har 9597 df-aleph 9980 |
| This theorem is referenced by: alephnbtwn 10111 alephnbtwn2 10112 alephordilem1 10113 alephord 10115 alephord2 10116 alephord3 10118 alephsucdom 10119 alephsuc2 10120 alephf1 10125 alephsdom 10126 alephdom2 10127 alephle 10128 cardaleph 10129 alephf1ALT 10143 alephfp 10148 dfac12k 10188 alephsing 10316 alephval2 10612 alephadd 10617 alephmul 10618 alephexp1 10619 alephsuc3 10620 alephreg 10622 pwcfsdom 10623 cfpwsdom 10624 gchaleph 10711 gchaleph2 10712 gch2 10715 minregex2 43548 alephiso2 43571 |
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