Theorem oasuc 3131

Description: Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56.

Assertion

Ref	Expression
oasuc	⊢ ((A ∈ On ∧ B ∈ On) → (A +o suc B) = suc (A +o B))

Proof of Theorem oasuc

Step	Hyp	Ref	Expression
1		rdgsuct 2983	. . 3 ⊢ (B ∈ On → (rec({⟨x, y⟩∣y = suc x}, A) ‘suc B) = ({⟨x, y⟩∣y = suc x} ‘(rec({⟨x, y⟩∣y = suc x}, A) ‘B)))
2	1	adantl 305	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (rec({⟨x, y⟩∣y = suc x}, A) ‘suc B) = ({⟨x, y⟩∣y = suc x} ‘(rec({⟨x, y⟩∣y = suc x}, A) ‘B)))
3		oav 3119	. . 3 ⊢ ((A ∈ On ∧ suc B ∈ On) → (A +o suc B) = (rec({⟨x, y⟩∣y = suc x}, A) ‘suc B))
4		suceloni 2314	. . 3 ⊢ (B ∈ On → suc B ∈ On)
5	3, 4	sylan2 346	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (A +o suc B) = (rec({⟨x, y⟩∣y = suc x}, A) ‘suc B))
6		oav 3119	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → (A +o B) = (rec({⟨x, y⟩∣y = suc x}, A) ‘B))
7	6	fveq2d 2836	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → ({⟨x, y⟩∣y = suc x} ‘(A +o B)) = ({⟨x, y⟩∣y = suc x} ‘(rec({⟨x, y⟩∣y = suc x}, A) ‘B)))
8		oprex 3018	. . . 4 ⊢ (A +o B) ∈ V
9	8	sucex 2303	. . . 4 ⊢ suc (A +o B) ∈ V
10		suceq 2288	. . . 4 ⊢ (x = (A +o B) → suc x = suc (A +o B))
11	8, 9, 10	fvopab 2877	. . 3 ⊢ ({⟨x, y⟩∣y = suc x} ‘(A +o B)) = suc (A +o B)
12	7, 11	syl5eqr 1138	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → suc (A +o B) = ({⟨x, y⟩∣y = suc x} ‘(rec({⟨x, y⟩∣y = suc x}, A) ‘B)))
13	2, 5, 12	3eqtr4d 1134	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A +o suc B) = suc (A +o B))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {copab 2055  Oncon0 2199  suc csuc 2201   ‘cfv 2422  reccrdg 2969  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  oa1suc 3132  oacl 3138  oa0r 3141  oaordi 3148  oawordri 3152  oawordeulem 3156  oalimcl 3162  oaass 3163  nnasuc 3168  nnacom 3175

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-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-opr 3003  df-oprab 3004  df-oadd 3106

metamath.org