Theorem oacl 4177

Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57.

Assertion

Ref	Expression
oacl	⊢ ((A ∈ On ⋀ B ∈ On) → (A +o B) ∈ On)

Proof of Theorem oacl

Step	Hyp	Ref	Expression
1		opreq2 3976	. . . 4 ⊢ (x = ∅ → (A +o x) = (A +o ∅))
2	1	eleq1d 1543	. . 3 ⊢ (x = ∅ → ((A +o x) ∈ On ↔ (A +o ∅) ∈ On))
3		opreq2 3976	. . . 4 ⊢ (x = y → (A +o x) = (A +o y))
4	3	eleq1d 1543	. . 3 ⊢ (x = y → ((A +o x) ∈ On ↔ (A +o y) ∈ On))
5		opreq2 3976	. . . 4 ⊢ (x = suc y → (A +o x) = (A +o suc y))
6	5	eleq1d 1543	. . 3 ⊢ (x = suc y → ((A +o x) ∈ On ↔ (A +o suc y) ∈ On))
7		opreq2 3976	. . . 4 ⊢ (x = B → (A +o x) = (A +o B))
8	7	eleq1d 1543	. . 3 ⊢ (x = B → ((A +o x) ∈ On ↔ (A +o B) ∈ On))
9		oa0 4162	. . . . 5 ⊢ (A ∈ On → (A +o ∅) = A)
10	9	eleq1d 1543	. . . 4 ⊢ (A ∈ On → ((A +o ∅) ∈ On ↔ A ∈ On))
11	10	ibir 595	. . 3 ⊢ (A ∈ On → (A +o ∅) ∈ On)
12		oasuc 4170	. . . . . 6 ⊢ ((A ∈ On ⋀ y ∈ On) → (A +o suc y) = suc (A +o y))
13	12	eleq1d 1543	. . . . 5 ⊢ ((A ∈ On ⋀ y ∈ On) → ((A +o suc y) ∈ On ↔ suc (A +o y) ∈ On))
14		suceloni 3069	. . . . 5 ⊢ ((A +o y) ∈ On → suc (A +o y) ∈ On)
15	13, 14	syl5bir 210	. . . 4 ⊢ ((A ∈ On ⋀ y ∈ On) → ((A +o y) ∈ On → (A +o suc y) ∈ On))
16	15	expcom 374	. . 3 ⊢ (y ∈ On → (A ∈ On → ((A +o y) ∈ On → (A +o suc y) ∈ On)))
17		visset 1816	. . . . . . 7 ⊢ x ∈ V
18		oalim 4174	. . . . . . 7 ⊢ ((A ∈ On ⋀ (x ∈ V ⋀ Lim x)) → (A +o x) = ∪y ∈ x (A +o y))
19	17, 18	mpanr1 711	. . . . . 6 ⊢ ((A ∈ On ⋀ Lim x) → (A +o x) = ∪y ∈ x (A +o y))
20	19	eleq1d 1543	. . . . 5 ⊢ ((A ∈ On ⋀ Lim x) → ((A +o x) ∈ On ↔ ∪y ∈ x (A +o y) ∈ On))
21		oprex 3990	. . . . . 6 ⊢ (A +o y) ∈ V
22	17, 21	iunon 3916	. . . . 5 ⊢ (∀y ∈ x (A +o y) ∈ On → ∪y ∈ x (A +o y) ∈ On)
23	20, 22	syl5bir 210	. . . 4 ⊢ ((A ∈ On ⋀ Lim x) → (∀y ∈ x (A +o y) ∈ On → (A +o x) ∈ On))
24	23	expcom 374	. . 3 ⊢ (Lim x → (A ∈ On → (∀y ∈ x (A +o y) ∈ On → (A +o x) ∈ On)))
25	2, 4, 6, 8, 11, 16, 24	tfinds3 3173	. 2 ⊢ (B ∈ On → (A ∈ On → (A +o B) ∈ On))
26	25	impcom 351	1 ⊢ ((A ∈ On ⋀ B ∈ On) → (A +o B) ∈ On)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∀wral 1648  Vcvv 1814  ∅c0 2284  ∪ciun 2571  Oncon0 2955  Lim wlim 2956  suc csuc 2957  (class class class)co 3970   +_o coa 4137

This theorem is referenced by:  omcl 4178  oaord 4188  oacan 4189  oaword 4190  oawordri 4191  oawordeulem 4195  oalimcl 4201  oaass 4202  odi 4217  oancom 4649

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2699  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786  ax-un 2873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2006  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-if 2367  df-pw 2407  df-sn 2417  df-pr 2418  df-tp 2420  df-op 2421  df-uni 2509  df-iun 2573  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-id 2842  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959  df-lim 2960  df-suc 2961  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-fv 3205  df-rdg 3939  df-opr 3972  df-oprab 3973  df-oadd 4142

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