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Theorem alephexp2 9981
Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 9979 (which works if the base is less than or equal to the exponent) and infmap 9976 (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.)
Assertion
Ref Expression
alephexp2 (𝐴 ∈ On → (2om (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
Distinct variable group:   𝑥,𝐴

Proof of Theorem alephexp2
StepHypRef Expression
1 alephgeom 9486 . . . 4 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
2 fvex 6659 . . . . 5 (ℵ‘𝐴) ∈ V
3 ssdomg 8533 . . . . 5 ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
42, 3ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
51, 4sylbi 219 . . 3 (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
6 domrefg 8522 . . . 4 ((ℵ‘𝐴) ∈ V → (ℵ‘𝐴) ≼ (ℵ‘𝐴))
72, 6ax-mp 5 . . 3 (ℵ‘𝐴) ≼ (ℵ‘𝐴)
8 infmap 9976 . . 3 ((ω ≼ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
95, 7, 8sylancl 588 . 2 (𝐴 ∈ On → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
10 pm3.2 472 . . . . 5 (𝐴 ∈ On → (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On)))
1110pm2.43i 52 . . . 4 (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On))
12 ssid 3968 . . . 4 𝐴𝐴
13 alephexp1 9979 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴𝐴) → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2om (ℵ‘𝐴)))
1411, 12, 13sylancl 588 . . 3 (𝐴 ∈ On → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2om (ℵ‘𝐴)))
15 enen1 8635 . . 3 (((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2om (ℵ‘𝐴)) → (((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (2om (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}))
1614, 15syl 17 . 2 (𝐴 ∈ On → (((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (2om (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}))
179, 16mpbid 234 1 (𝐴 ∈ On → (2om (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2114  {cab 2798  Vcvv 3473  wss 3913   class class class wbr 5042  Oncon0 6167  cfv 6331  (class class class)co 7133  ωcom 7558  2oc2o 8074  m cmap 8384  cen 8484  cdom 8485  cale 9343
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-ac2 9863
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-oi 8952  df-har 9000  df-card 9346  df-aleph 9347  df-acn 9349  df-ac 9520
This theorem is referenced by:  gch-kn  10077
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