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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 10106 | . . 3 ⊢ (ℵ‘∅) = ω | |
| 2 | 0ss 4400 | . . . 4 ⊢ ∅ ⊆ 𝐴 | |
| 3 | 0elon 6438 | . . . . 5 ⊢ ∅ ∈ On | |
| 4 | alephord3 10118 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (ℵ‘∅) ⊆ (ℵ‘𝐴))) | |
| 5 | 3, 4 | mpan 690 | . . . 4 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (ℵ‘∅) ⊆ (ℵ‘𝐴))) |
| 6 | 2, 5 | mpbii 233 | . . 3 ⊢ (𝐴 ∈ On → (ℵ‘∅) ⊆ (ℵ‘𝐴)) |
| 7 | 1, 6 | eqsstrrid 4023 | . 2 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴)) |
| 8 | peano1 7910 | . . . . . 6 ⊢ ∅ ∈ ω | |
| 9 | ordom 7897 | . . . . . . . 8 ⊢ Ord ω | |
| 10 | ord0 6437 | . . . . . . . 8 ⊢ Ord ∅ | |
| 11 | ordtri1 6417 | . . . . . . . 8 ⊢ ((Ord ω ∧ Ord ∅) → (ω ⊆ ∅ ↔ ¬ ∅ ∈ ω)) | |
| 12 | 9, 10, 11 | mp2an 692 | . . . . . . 7 ⊢ (ω ⊆ ∅ ↔ ¬ ∅ ∈ ω) |
| 13 | 12 | con2bii 357 | . . . . . 6 ⊢ (∅ ∈ ω ↔ ¬ ω ⊆ ∅) |
| 14 | 8, 13 | mpbi 230 | . . . . 5 ⊢ ¬ ω ⊆ ∅ |
| 15 | ndmfv 6941 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅) | |
| 16 | 15 | sseq2d 4016 | . . . . 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 10105 | . . . 4 ⊢ ℵ Fn On | |
| 20 | 19 | fndmi 6672 | . . 3 ⊢ dom ℵ = On |
| 21 | 18, 20 | eleqtrdi 2851 | . 2 ⊢ (ω ⊆ (ℵ‘𝐴) → 𝐴 ∈ On) |
| 22 | 7, 21 | impbii 209 | 1 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∈ wcel 2108 ⊆ wss 3951 ∅c0 4333 dom cdm 5685 Ord word 6383 Oncon0 6384 ‘cfv 6561 ωcom 7887 ℵcale 9976 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-oi 9550 df-har 9597 df-card 9979 df-aleph 9980 |
| This theorem is referenced by: alephislim 10123 cardalephex 10130 isinfcard 10132 alephval3 10150 alephval2 10612 alephadd 10617 alephmul 10618 alephexp1 10619 alephsuc3 10620 alephexp2 10621 alephreg 10622 pwcfsdom 10623 cfpwsdom 10624 gchaleph 10711 gchaleph2 10712 |
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