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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 8771 | . . 3 ⊢ ℵ Fn On | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅)) | |
3 | 2 | eleq1d 2672 | . . . . 5 ⊢ (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On)) |
4 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦)) | |
5 | 4 | eleq1d 2672 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On)) |
6 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦)) | |
7 | 6 | eleq1d 2672 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On)) |
8 | aleph0 8772 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
9 | omelon 8426 | . . . . . 6 ⊢ ω ∈ On | |
10 | 8, 9 | eqeltri 2684 | . . . . 5 ⊢ (ℵ‘∅) ∈ On |
11 | alephsuc 8774 | . . . . . . 7 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦))) | |
12 | harcl 8349 | . . . . . . 7 ⊢ (har‘(ℵ‘𝑦)) ∈ On | |
13 | 11, 12 | syl6eqel 2696 | . . . . . 6 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) ∈ On) |
14 | 13 | a1d 25 | . . . . 5 ⊢ (𝑦 ∈ On → ((ℵ‘𝑦) ∈ On → (ℵ‘suc 𝑦) ∈ On)) |
15 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
16 | iunon 7323 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) | |
17 | 15, 16 | mpan 702 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On) |
18 | alephlim 8773 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) | |
19 | 15, 18 | mpan 702 | . . . . . . 7 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦)) |
20 | 19 | eleq1d 2672 | . . . . . 6 ⊢ (Lim 𝑥 → ((ℵ‘𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On)) |
21 | 17, 20 | syl5ibr 235 | . . . . 5 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ On → (ℵ‘𝑥) ∈ On)) |
22 | 3, 5, 7, 5, 10, 14, 21 | tfinds 6951 | . . . 4 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ On) |
23 | 22 | rgen 2906 | . . 3 ⊢ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On |
24 | ffnfv 6295 | . . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On)) | |
25 | 1, 23, 24 | mpbir2an 957 | . 2 ⊢ ℵ:On⟶On |
26 | 0elon 5695 | . 2 ⊢ ∅ ∈ On | |
27 | 25, 26 | f0cli 6278 | 1 ⊢ (ℵ‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∅c0 3874 ∪ ciun 4455 Oncon0 5640 Lim wlim 5641 suc csuc 5642 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 ωcom 6957 harchar 8344 ℵcale 8645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-en 7842 df-dom 7843 df-oi 8298 df-har 8346 df-aleph 8649 |
This theorem is referenced by: alephnbtwn 8777 alephnbtwn2 8778 alephordilem1 8779 alephord 8781 alephord2 8782 alephord3 8784 alephsucdom 8785 alephsuc2 8786 alephf1 8791 alephsdom 8792 alephdom2 8793 alephle 8794 cardaleph 8795 alephf1ALT 8809 alephfp 8814 dfac12k 8852 alephsing 8981 alephval2 9273 alephadd 9278 alephmul 9279 alephexp1 9280 alephsuc3 9281 alephreg 9283 pwcfsdom 9284 cfpwsdom 9285 gchaleph 9372 gchaleph2 9373 gch2 9376 |
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