Theorem alephexp2 10268

Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 10266 (which works if the base is less than or equal to the exponent) and infmap 10263 (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 9769	. . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
2		fvex 6769	. . . . 5 ⊢ (ℵ‘𝐴) ∈ V
3		ssdomg 8741	. . . . 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 216	. . 3 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
6		domrefg 8730	. . . 4 ⊢ ((ℵ‘𝐴) ∈ V → (ℵ‘𝐴) ≼ (ℵ‘𝐴))
7	2, 6	ax-mp 5	. . 3 ⊢ (ℵ‘𝐴) ≼ (ℵ‘𝐴)
8		infmap 10263	. . 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 585	. 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
10		pm3.2 469	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On)))
11	10	pm2.43i 52	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ∧ 𝐴 ∈ On))
12		ssid 3939	. . . 4 ⊢ 𝐴 ⊆ 𝐴
13		alephexp1 10266	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴 ⊆ 𝐴) → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2o ↑m (ℵ‘𝐴)))
14	11, 12, 13	sylancl 585	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2o ↑m (ℵ‘𝐴)))
15		enen1 8853	. . 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 231	1 ⊢ (𝐴 ∈ On → (2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  {cab 2715  Vcvv 3422   ⊆ wss 3883   class class class wbr 5070  Oncon0 6251  ‘cfv 6418  (class class class)co 7255  ωcom 7687  2_oc2o 8261   ↑_m cmap 8573   ≈ cen 8688   ≼ cdom 8689  ℵcale 9625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-ac2 10150

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-har 9246  df-card 9628  df-aleph 9629  df-acn 9631  df-ac 9803

This theorem is referenced by:  gch-kn  10364