Theorem alephexp2 10618

Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 10616 (which works if the base is less than or equal to the exponent) and infmap 10613 (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 10119	. . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
2		fvex 6919	. . . . 5 ⊢ (ℵ‘𝐴) ∈ V
3		ssdomg 9038	. . . . 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 217	. . 3 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
6		domrefg 9025	. . . 4 ⊢ ((ℵ‘𝐴) ∈ V → (ℵ‘𝐴) ≼ (ℵ‘𝐴))
7	2, 6	ax-mp 5	. . 3 ⊢ (ℵ‘𝐴) ≼ (ℵ‘𝐴)
8		infmap 10613	. . 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 586	. 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 4017	. . . 4 ⊢ 𝐴 ⊆ 𝐴
13		alephexp1 10616	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐴 ⊆ 𝐴) → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2o ↑m (ℵ‘𝐴)))
14	11, 12, 13	sylancl 586	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ↑m (ℵ‘𝐴)) ≈ (2o ↑m (ℵ‘𝐴)))
15		enen1 9155	. . 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 232	1 ⊢ (𝐴 ∈ On → (2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2105  {cab 2711  Vcvv 3477   ⊆ wss 3962   class class class wbr 5147  Oncon0 6385  ‘cfv 6562  (class class class)co 7430  ωcom 7886  2_oc2o 8498   ↑_m cmap 8864   ≈ cen 8980   ≼ cdom 8981  ℵcale 9973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-ac2 10500

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-har 9594  df-card 9976  df-aleph 9977  df-acn 9979  df-ac 10153

This theorem is referenced by:  gch-kn  10714