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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 10602 (which works if the base is less than or equal to the exponent) and infmap 10599 (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 10105 | . . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) | |
2 | fvex 6905 | . . . . 5 ⊢ (ℵ‘𝐴) ∈ V | |
3 | ssdomg 9019 | . . . . 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 216 | . . 3 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴)) |
6 | domrefg 9006 | . . . 4 ⊢ ((ℵ‘𝐴) ∈ V → (ℵ‘𝐴) ≼ (ℵ‘𝐴)) | |
7 | 2, 6 | ax-mp 5 | . . 3 ⊢ (ℵ‘𝐴) ≼ (ℵ‘𝐴) |
8 | infmap 10599 | . . 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 584 | . 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}) |
10 | pm3.2 468 | . . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On))) | |
11 | 10 | pm2.43i 52 | . . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On)) |
12 | ssid 3995 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
13 | alephexp1 10602 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴 ⊆ 𝐴) → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2o ↑m (ℵ‘𝐴))) | |
14 | 11, 12, 13 | sylancl 584 | . . 3 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2o ↑m (ℵ‘𝐴))) |
15 | enen1 9140 | . . 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 231 | 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 205 ∧ wa 394 ∈ wcel 2098 {cab 2702 Vcvv 3463 ⊆ wss 3939 class class class wbr 5143 Oncon0 6364 ‘cfv 6543 (class class class)co 7416 ωcom 7868 2oc2o 8479 ↑m cmap 8843 ≈ cen 8959 ≼ cdom 8960 ℵcale 9959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-ac2 10486 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-oi 9533 df-har 9580 df-card 9962 df-aleph 9963 df-acn 9965 df-ac 10139 |
This theorem is referenced by: gch-kn 10700 |
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