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Mirrors > Home > MPE Home > Th. List > alephexp2 | Structured version Visualization version GIF version |
Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 9736 (which works if the base is less than or equal to the exponent) and infmap 9733 (which works if the exponent is less than or equal to the base). They can be equated only when the base is equal to the exponent, and this is the result. (Contributed by NM, 23-Oct-2004.) |
Ref | Expression |
---|---|
alephexp2 | ⊢ (𝐴 ∈ On → (2o ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephgeom 9238 | . . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) | |
2 | fvex 6459 | . . . . 5 ⊢ (ℵ‘𝐴) ∈ V | |
3 | ssdomg 8287 | . . . . 5 ⊢ ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)) |
5 | 1, 4 | sylbi 209 | . . 3 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴)) |
6 | domrefg 8276 | . . . 4 ⊢ ((ℵ‘𝐴) ∈ V → (ℵ‘𝐴) ≼ (ℵ‘𝐴)) | |
7 | 2, 6 | ax-mp 5 | . . 3 ⊢ (ℵ‘𝐴) ≼ (ℵ‘𝐴) |
8 | infmap 9733 | . . 3 ⊢ ((ω ≼ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}) | |
9 | 5, 7, 8 | sylancl 580 | . 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}) |
10 | pm3.2 463 | . . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On))) | |
11 | 10 | pm2.43i 52 | . . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On)) |
12 | ssid 3841 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
13 | alephexp1 9736 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴 ⊆ 𝐴) → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ (2o ↑𝑚 (ℵ‘𝐴))) | |
14 | 11, 12, 13 | sylancl 580 | . . 3 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ (2o ↑𝑚 (ℵ‘𝐴))) |
15 | enen1 8388 | . . 3 ⊢ (((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ (2o ↑𝑚 (ℵ‘𝐴)) → (((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (2o ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (2o ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})) |
17 | 9, 16 | mpbid 224 | 1 ⊢ (𝐴 ∈ On → (2o ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2106 {cab 2762 Vcvv 3397 ⊆ wss 3791 class class class wbr 4886 Oncon0 5976 ‘cfv 6135 (class class class)co 6922 ωcom 7343 2oc2o 7837 ↑𝑚 cmap 8140 ≈ cen 8238 ≼ cdom 8239 ℵcale 9095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-ac2 9620 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-oi 8704 df-har 8752 df-card 9098 df-aleph 9099 df-acn 9101 df-ac 9272 |
This theorem is referenced by: gch-kn 9834 |
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