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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 9480 | . . 3 ⊢ (ℵ‘∅) = ω | |
2 | 0ss 4347 | . . . 4 ⊢ ∅ ⊆ 𝐴 | |
3 | 0elon 6237 | . . . . 5 ⊢ ∅ ∈ On | |
4 | alephord3 9492 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (ℵ‘∅) ⊆ (ℵ‘𝐴))) | |
5 | 3, 4 | mpan 686 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (ℵ‘∅) ⊆ (ℵ‘𝐴))) |
6 | 2, 5 | mpbii 234 | . . 3 ⊢ (𝐴 ∈ On → (ℵ‘∅) ⊆ (ℵ‘𝐴)) |
7 | 1, 6 | eqsstrrid 4013 | . 2 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴)) |
8 | peano1 7590 | . . . . . 6 ⊢ ∅ ∈ ω | |
9 | ordom 7578 | . . . . . . . 8 ⊢ Ord ω | |
10 | ord0 6236 | . . . . . . . 8 ⊢ Ord ∅ | |
11 | ordtri1 6217 | . . . . . . . 8 ⊢ ((Ord ω ∧ Ord ∅) → (ω ⊆ ∅ ↔ ¬ ∅ ∈ ω)) | |
12 | 9, 10, 11 | mp2an 688 | . . . . . . 7 ⊢ (ω ⊆ ∅ ↔ ¬ ∅ ∈ ω) |
13 | 12 | con2bii 359 | . . . . . 6 ⊢ (∅ ∈ ω ↔ ¬ ω ⊆ ∅) |
14 | 8, 13 | mpbi 231 | . . . . 5 ⊢ ¬ ω ⊆ ∅ |
15 | ndmfv 6693 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅) | |
16 | 15 | sseq2d 3996 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ → (ω ⊆ (ℵ‘𝐴) ↔ ω ⊆ ∅)) |
17 | 14, 16 | mtbiri 328 | . . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → ¬ ω ⊆ (ℵ‘𝐴)) |
18 | 17 | con4i 114 | . . 3 ⊢ (ω ⊆ (ℵ‘𝐴) → 𝐴 ∈ dom ℵ) |
19 | alephfnon 9479 | . . . 4 ⊢ ℵ Fn On | |
20 | fndm 6448 | . . . 4 ⊢ (ℵ Fn On → dom ℵ = On) | |
21 | 19, 20 | ax-mp 5 | . . 3 ⊢ dom ℵ = On |
22 | 18, 21 | eleqtrdi 2920 | . 2 ⊢ (ω ⊆ (ℵ‘𝐴) → 𝐴 ∈ On) |
23 | 7, 22 | impbii 210 | 1 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ∅c0 4288 dom cdm 5548 Ord word 6183 Oncon0 6184 Fn wfn 6343 ‘cfv 6348 ωcom 7569 ℵcale 9353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-oi 8962 df-har 9010 df-card 9356 df-aleph 9357 |
This theorem is referenced by: alephislim 9497 cardalephex 9504 isinfcard 9506 alephval3 9524 alephval2 9982 alephadd 9987 alephmul 9988 alephexp1 9989 alephsuc3 9990 alephexp2 9991 alephreg 9992 pwcfsdom 9993 cfpwsdom 9994 gchaleph 10081 gchaleph2 10082 |
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