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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 9208 | . . 3 ⊢ ℵ Fn On | |
2 | fveq2 6437 | . . . . . 6 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅)) | |
3 | 2 | eleq1d 2891 | . . . . 5 ⊢ (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On)) |
4 | fveq2 6437 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦)) | |
5 | 4 | eleq1d 2891 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On)) |
6 | fveq2 6437 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦)) | |
7 | 6 | eleq1d 2891 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On)) |
8 | aleph0 9209 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
9 | omelon 8827 | . . . . . 6 ⊢ ω ∈ On | |
10 | 8, 9 | eqeltri 2902 | . . . . 5 ⊢ (ℵ‘∅) ∈ On |
11 | alephsuc 9211 | . . . . . . 7 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦))) | |
12 | harcl 8742 | . . . . . . 7 ⊢ (har‘(ℵ‘𝑦)) ∈ On | |
13 | 11, 12 | syl6eqel 2914 | . . . . . 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 7707 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) | |
17 | 15, 16 | mpan 681 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) |
18 | alephlim 9210 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) | |
19 | 15, 18 | mpan 681 | . . . . . . 7 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) |
20 | 19 | eleq1d 2891 | . . . . . 6 ⊢ (Lim 𝑥 → ((ℵ‘𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On)) |
21 | 17, 20 | syl5ibr 238 | . . . . 5 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → (ℵ‘𝑥) ∈ On)) |
22 | 3, 5, 7, 5, 10, 14, 21 | tfinds 7325 | . . . 4 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ On) |
23 | 22 | rgen 3131 | . . 3 ⊢ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On |
24 | ffnfv 6642 | . . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On)) | |
25 | 1, 23, 24 | mpbir2an 702 | . 2 ⊢ ℵ:On⟶On |
26 | 0elon 6020 | . 2 ⊢ ∅ ∈ On | |
27 | 25, 26 | f0cli 6624 | 1 ⊢ (ℵ‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 ∀wral 3117 Vcvv 3414 ∅c0 4146 ∪ ciun 4742 Oncon0 5967 Lim wlim 5968 suc csuc 5969 Fn wfn 6122 ⟶wf 6123 ‘cfv 6127 ωcom 7331 harchar 8737 ℵcale 9082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-en 8229 df-dom 8230 df-oi 8691 df-har 8739 df-aleph 9086 |
This theorem is referenced by: alephnbtwn 9214 alephnbtwn2 9215 alephordilem1 9216 alephord 9218 alephord2 9219 alephord3 9221 alephsucdom 9222 alephsuc2 9223 alephf1 9228 alephsdom 9229 alephdom2 9230 alephle 9231 cardaleph 9232 alephf1ALT 9246 alephfp 9251 dfac12k 9291 alephsing 9420 alephval2 9716 alephadd 9721 alephmul 9722 alephexp1 9723 alephsuc3 9724 alephreg 9726 pwcfsdom 9727 cfpwsdom 9728 gchaleph 9815 gchaleph2 9816 gch2 9819 |
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