Theorem alephlim 3670

Description: Value of the aleph function at a limit ordinal. Definition 12(iii) of [Suppes] p. 91.

Assertion

Ref	Expression
alephlim	⊢ ((A ∈ B ∧ Lim A) → (ℵ ‘A) = ∪x ∈ A (ℵ ‘x))

Distinct variable group(s):   x,A

Proof of Theorem alephlim

Step	Hyp	Ref	Expression
1		rdglim2a 2988	. 2 ⊢ ((A ∈ B ∧ Lim A) → (rec({⟨y, z⟩∣z = ∩{w ∈ On∣y ≺ w}}, ω) ‘A) = ∪x ∈ A (rec({⟨y, z⟩∣z = ∩{w ∈ On∣y ≺ w}}, ω) ‘x))
2		df-aleph 3624	. . 3 ⊢ ℵ = rec({⟨y, z⟩∣z = ∩{w ∈ On∣y ≺ w}}, ω)
3	2	fveq1i 2833	. 2 ⊢ (ℵ ‘A) = (rec({⟨y, z⟩∣z = ∩{w ∈ On∣y ≺ w}}, ω) ‘A)
4	2	fveq1i 2833	. . . 4 ⊢ (ℵ ‘x) = (rec({⟨y, z⟩∣z = ∩{w ∈ On∣y ≺ w}}, ω) ‘x)
5	4	a1i 7	. . 3 ⊢ (x ∈ A → (ℵ ‘x) = (rec({⟨y, z⟩∣z = ∩{w ∈ On∣y ≺ w}}, ω) ‘x))
6	5	iuneq2i 2008	. 2 ⊢ ∪x ∈ A (ℵ ‘x) = ∪x ∈ A (rec({⟨y, z⟩∣z = ∩{w ∈ On∣y ≺ w}}, ω) ‘x)
7	1, 3, 6	3eqtr4g 1147	1 ⊢ ((A ∈ B ∧ Lim A) → (ℵ ‘A) = ∪x ∈ A (ℵ ‘x))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {crab 1204  ∩cint 1965  ∪ciun 1994   class class class wbr 2054  {copab 2055  Oncon0 2199  Lim wlim 2200  ωcom 2372   ‘cfv 2422  reccrdg 2969   ≺ csdm 3273  ℵcale 3621

This theorem is referenced by:  alephon 3671  alephcard 3673  alephordlem2 3678  alephordi 3679  cardaleph 3690

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-aleph 3624

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