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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 10065 | . . 3 ⊢ (ℵ‘∅) = ω | |
2 | 0ss 4396 | . . . 4 ⊢ ∅ ⊆ 𝐴 | |
3 | 0elon 6418 | . . . . 5 ⊢ ∅ ∈ On | |
4 | alephord3 10077 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (ℵ‘∅) ⊆ (ℵ‘𝐴))) | |
5 | 3, 4 | mpan 687 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (ℵ‘∅) ⊆ (ℵ‘𝐴))) |
6 | 2, 5 | mpbii 232 | . . 3 ⊢ (𝐴 ∈ On → (ℵ‘∅) ⊆ (ℵ‘𝐴)) |
7 | 1, 6 | eqsstrrid 4031 | . 2 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴)) |
8 | peano1 7883 | . . . . . 6 ⊢ ∅ ∈ ω | |
9 | ordom 7869 | . . . . . . . 8 ⊢ Ord ω | |
10 | ord0 6417 | . . . . . . . 8 ⊢ Ord ∅ | |
11 | ordtri1 6397 | . . . . . . . 8 ⊢ ((Ord ω ∧ Ord ∅) → (ω ⊆ ∅ ↔ ¬ ∅ ∈ ω)) | |
12 | 9, 10, 11 | mp2an 689 | . . . . . . 7 ⊢ (ω ⊆ ∅ ↔ ¬ ∅ ∈ ω) |
13 | 12 | con2bii 357 | . . . . . 6 ⊢ (∅ ∈ ω ↔ ¬ ω ⊆ ∅) |
14 | 8, 13 | mpbi 229 | . . . . 5 ⊢ ¬ ω ⊆ ∅ |
15 | ndmfv 6926 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅) | |
16 | 15 | sseq2d 4014 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ → (ω ⊆ (ℵ‘𝐴) ↔ ω ⊆ ∅)) |
17 | 14, 16 | mtbiri 327 | . . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → ¬ ω ⊆ (ℵ‘𝐴)) |
18 | 17 | con4i 114 | . . 3 ⊢ (ω ⊆ (ℵ‘𝐴) → 𝐴 ∈ dom ℵ) |
19 | alephfnon 10064 | . . . 4 ⊢ ℵ Fn On | |
20 | 19 | fndmi 6653 | . . 3 ⊢ dom ℵ = On |
21 | 18, 20 | eleqtrdi 2842 | . 2 ⊢ (ω ⊆ (ℵ‘𝐴) → 𝐴 ∈ On) |
22 | 7, 21 | impbii 208 | 1 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2105 ⊆ wss 3948 ∅c0 4322 dom cdm 5676 Ord word 6363 Oncon0 6364 ‘cfv 6543 ωcom 7859 ℵcale 9935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-oi 9509 df-har 9556 df-card 9938 df-aleph 9939 |
This theorem is referenced by: alephislim 10082 cardalephex 10089 isinfcard 10091 alephval3 10109 alephval2 10571 alephadd 10576 alephmul 10577 alephexp1 10578 alephsuc3 10579 alephexp2 10580 alephreg 10581 pwcfsdom 10582 cfpwsdom 10583 gchaleph 10670 gchaleph2 10671 |
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