Theorem alephexp2 9738

Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 9736 (which works if the base is less than or equal to the exponent) and infmap 9733 (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 9238	. . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
2		fvex 6459	. . . . 5 ⊢ (ℵ‘𝐴) ∈ V
3		ssdomg 8287	. . . . 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 209	. . 3 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
6		domrefg 8276	. . . 4 ⊢ ((ℵ‘𝐴) ∈ V → (ℵ‘𝐴) ≼ (ℵ‘𝐴))
7	2, 6	ax-mp 5	. . 3 ⊢ (ℵ‘𝐴) ≼ (ℵ‘𝐴)
8		infmap 9733	. . 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 580	. 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)))
11	10	pm2.43i 52	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On))
12		ssid 3841	. . . 4 ⊢ 𝐴 ⊆ 𝐴
13		alephexp1 9736	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴 ⊆ 𝐴) → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ (2o ↑𝑚 (ℵ‘𝐴)))
14	11, 12, 13	sylancl 580	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)) ≈ (2o ↑𝑚 (ℵ‘𝐴)))
15		enen1 8388	. . 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 224	1 ⊢ (𝐴 ∈ On → (2o ↑𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2106  {cab 2762  Vcvv 3397   ⊆ wss 3791   class class class wbr 4886  Oncon0 5976  ‘cfv 6135  (class class class)co 6922  ωcom 7343  2_oc2o 7837   ↑_𝑚 cmap 8140   ≈ cen 8238   ≼ cdom 8239  ℵcale 9095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-ac2 9620

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-oi 8704  df-har 8752  df-card 9098  df-aleph 9099  df-acn 9101  df-ac 9272

This theorem is referenced by:  gch-kn  9834