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Theorem alephexp2 9388
Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 9386 (which works if the base is less than or equal to the exponent) and infmap 9383 (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 8890 . . . 4 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
2 fvex 6188 . . . . 5 (ℵ‘𝐴) ∈ V
3 ssdomg 7986 . . . . 5 ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
42, 3ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
51, 4sylbi 207 . . 3 (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
6 domrefg 7975 . . . 4 ((ℵ‘𝐴) ∈ V → (ℵ‘𝐴) ≼ (ℵ‘𝐴))
72, 6ax-mp 5 . . 3 (ℵ‘𝐴) ≼ (ℵ‘𝐴)
8 infmap 9383 . . 3 ((ω ≼ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
95, 7, 8sylancl 693 . 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)))
1110pm2.43i 52 . . . 4 (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On))
12 ssid 3616 . . . 4 𝐴𝐴
13 alephexp1 9386 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴𝐴) → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ (2𝑜𝑚 (ℵ‘𝐴)))
1411, 12, 13sylancl 693 . . 3 (𝐴 ∈ On → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ (2𝑜𝑚 (ℵ‘𝐴)))
15 enen1 8085 . . 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 222 1 (𝐴 ∈ On → (2𝑜𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1988  {cab 2606  Vcvv 3195  wss 3567   class class class wbr 4644  Oncon0 5711  cfv 5876  (class class class)co 6635  ωcom 7050  2𝑜c2o 7539  𝑚 cmap 7842  cen 7937  cdom 7938  cale 8747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-har 8448  df-card 8750  df-aleph 8751  df-acn 8753  df-ac 8924
This theorem is referenced by:  gch-kn  9484
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