![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > alephgeom | Structured version Visualization version GIF version |
Description: Every aleph is greater than or equal to the set of natural numbers. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
alephgeom | ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aleph0 9089 | . . 3 ⊢ (ℵ‘∅) = ω | |
2 | 0ss 4116 | . . . 4 ⊢ ∅ ⊆ 𝐴 | |
3 | 0elon 5921 | . . . . 5 ⊢ ∅ ∈ On | |
4 | alephord3 9101 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (ℵ‘∅) ⊆ (ℵ‘𝐴))) | |
5 | 3, 4 | mpan 662 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (ℵ‘∅) ⊆ (ℵ‘𝐴))) |
6 | 2, 5 | mpbii 223 | . . 3 ⊢ (𝐴 ∈ On → (ℵ‘∅) ⊆ (ℵ‘𝐴)) |
7 | 1, 6 | syl5eqssr 3799 | . 2 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴)) |
8 | peano1 7232 | . . . . . 6 ⊢ ∅ ∈ ω | |
9 | ordom 7221 | . . . . . . . 8 ⊢ Ord ω | |
10 | ord0 5920 | . . . . . . . 8 ⊢ Ord ∅ | |
11 | ordtri1 5899 | . . . . . . . 8 ⊢ ((Ord ω ∧ Ord ∅) → (ω ⊆ ∅ ↔ ¬ ∅ ∈ ω)) | |
12 | 9, 10, 11 | mp2an 664 | . . . . . . 7 ⊢ (ω ⊆ ∅ ↔ ¬ ∅ ∈ ω) |
13 | 12 | con2bii 346 | . . . . . 6 ⊢ (∅ ∈ ω ↔ ¬ ω ⊆ ∅) |
14 | 8, 13 | mpbi 220 | . . . . 5 ⊢ ¬ ω ⊆ ∅ |
15 | ndmfv 6359 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅) | |
16 | 15 | sseq2d 3782 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ → (ω ⊆ (ℵ‘𝐴) ↔ ω ⊆ ∅)) |
17 | 14, 16 | mtbiri 316 | . . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → ¬ ω ⊆ (ℵ‘𝐴)) |
18 | 17 | con4i 114 | . . 3 ⊢ (ω ⊆ (ℵ‘𝐴) → 𝐴 ∈ dom ℵ) |
19 | alephfnon 9088 | . . . 4 ⊢ ℵ Fn On | |
20 | fndm 6130 | . . . 4 ⊢ (ℵ Fn On → dom ℵ = On) | |
21 | 19, 20 | ax-mp 5 | . . 3 ⊢ dom ℵ = On |
22 | 18, 21 | syl6eleq 2860 | . 2 ⊢ (ω ⊆ (ℵ‘𝐴) → 𝐴 ∈ On) |
23 | 7, 22 | impbii 199 | 1 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ∅c0 4063 dom cdm 5249 Ord word 5865 Oncon0 5866 Fn wfn 6026 ‘cfv 6031 ωcom 7212 ℵcale 8962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-oi 8571 df-har 8619 df-card 8965 df-aleph 8966 |
This theorem is referenced by: alephislim 9106 cardalephex 9113 isinfcard 9115 alephval3 9133 alephval2 9596 alephadd 9601 alephmul 9602 alephexp1 9603 alephsuc3 9604 alephexp2 9605 alephreg 9606 pwcfsdom 9607 cfpwsdom 9608 gchaleph 9695 gchaleph2 9696 |
Copyright terms: Public domain | W3C validator |