Theorem alephexp2 10666

Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 10664 (which works if the base is less than or equal to the exponent) and infmap 10661 (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 → (2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})

Distinct variable group:   𝑥,𝐴

Proof of Theorem alephexp2

Step	Hyp	Ref	Expression
1		alephgeom 10161	. . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
2		fvex 6898	. . . . 5 ⊢ (ℵ‘𝐴) ∈ V
3		ssdomg 9027	. . . . 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 220	. . 3 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
6		domrefg 9014	. . . 4 ⊢ ((ℵ‘𝐴) ∈ V → (ℵ‘𝐴) ≼ (ℵ‘𝐴))
7	2, 6	ax-mp 5	. . 3 ⊢ (ℵ‘𝐴) ≼ (ℵ‘𝐴)
8		infmap 10661	. . 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 598	. 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
10		pm3.2 475	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On)))
11	10	pm2.43i 53	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On))
12		ssid 3930	. . . 4 ⊢ 𝐴 ⊆ 𝐴
13		alephexp1 10664	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴 ⊆ 𝐴) → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2o ↑m (ℵ‘𝐴)))
14	11, 12, 13	sylancl 598	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2o ↑m (ℵ‘𝐴)))
15		enen1 9136	. . 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 18	. 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 235	1 ⊢ (𝐴 ∈ On → (2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  {cab 2738  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102  Oncon0 6362  ‘cfv 6538  (class class class)co 7420  ωcom 7877  2_oc2o 8470   ↑_m cmap 8847   ≈ cen 8970   ≼ cdom 8971  ℵcale 10017

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9642  ax-ac2 10541

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-oi 9504  df-har 9551  df-card 10020  df-aleph 10021  df-acn 10023  df-ac 10195

This theorem is used by:  gch-kn  10762