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Theorem alephexp2 9595
Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 9593 (which works if the base is less than or equal to the exponent) and infmap 9590 (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 9095 . . . 4 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
2 fvex 6362 . . . . 5 (ℵ‘𝐴) ∈ V
3 ssdomg 8167 . . . . 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 8156 . . . 4 ((ℵ‘𝐴) ∈ V → (ℵ‘𝐴) ≼ (ℵ‘𝐴))
72, 6ax-mp 5 . . 3 (ℵ‘𝐴) ≼ (ℵ‘𝐴)
8 infmap 9590 . . 3 ((ω ≼ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
95, 7, 8sylancl 697 . 2 (𝐴 ∈ On → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
10 pm3.2 462 . . . . 5 (𝐴 ∈ On → (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On)))
1110pm2.43i 52 . . . 4 (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On))
12 ssid 3765 . . . 4 𝐴𝐴
13 alephexp1 9593 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴𝐴) → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ (2𝑜𝑚 (ℵ‘𝐴)))
1411, 12, 13sylancl 697 . . 3 (𝐴 ∈ On → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ (2𝑜𝑚 (ℵ‘𝐴)))
15 enen1 8265 . . 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 383  wcel 2139  {cab 2746  Vcvv 3340  wss 3715   class class class wbr 4804  Oncon0 5884  cfv 6049  (class class class)co 6813  ωcom 7230  2𝑜c2o 7723  𝑚 cmap 8023  cen 8118  cdom 8119  cale 8952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-ac2 9477
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-oi 8580  df-har 8628  df-card 8955  df-aleph 8956  df-acn 8958  df-ac 9129
This theorem is referenced by:  gch-kn  9691
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