Theorem oa0r 3141

Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57.

Assertion

Ref	Expression
oa0r	⊢ (A ∈ On → (∅ +o A) = A)

Proof of Theorem oa0r

Step	Hyp	Ref	Expression
1		opreq2 3007	. . 3 ⊢ (x = ∅ → (∅ +o x) = (∅ +o ∅))
2		id 9	. . 3 ⊢ (x = ∅ → x = ∅)
3	1, 2	cleq12d 1115	. 2 ⊢ (x = ∅ → ((∅ +o x) = x ↔ (∅ +o ∅) = ∅))
4		opreq2 3007	. . 3 ⊢ (x = y → (∅ +o x) = (∅ +o y))
5		id 9	. . 3 ⊢ (x = y → x = y)
6	4, 5	cleq12d 1115	. 2 ⊢ (x = y → ((∅ +o x) = x ↔ (∅ +o y) = y))
7		opreq2 3007	. . 3 ⊢ (x = suc y → (∅ +o x) = (∅ +o suc y))
8		id 9	. . 3 ⊢ (x = suc y → x = suc y)
9	7, 8	cleq12d 1115	. 2 ⊢ (x = suc y → ((∅ +o x) = x ↔ (∅ +o suc y) = suc y))
10		opreq2 3007	. . 3 ⊢ (x = A → (∅ +o x) = (∅ +o A))
11		id 9	. . 3 ⊢ (x = A → x = A)
12	10, 11	cleq12d 1115	. 2 ⊢ (x = A → ((∅ +o x) = x ↔ (∅ +o A) = A))
13		0elon 2277	. . 3 ⊢ ∅ ∈ On
14		oa0 3124	. . 3 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
15	13, 14	ax-mp 6	. 2 ⊢ (∅ +o ∅) = ∅
16		oasuc 3131	. . . . 5 ⊢ ((∅ ∈ On ∧ y ∈ On) → (∅ +o suc y) = suc (∅ +o y))
17	13, 16	mpan 518	. . . 4 ⊢ (y ∈ On → (∅ +o suc y) = suc (∅ +o y))
18		suceq 2288	. . . 4 ⊢ ((∅ +o y) = y → suc (∅ +o y) = suc y)
19	17, 18	sylan9eq 1144	. . 3 ⊢ ((y ∈ On ∧ (∅ +o y) = y) → (∅ +o suc y) = suc y)
20	19	exp 291	. 2 ⊢ (y ∈ On → ((∅ +o y) = y → (∅ +o suc y) = suc y))
21		visset 1350	. . . . 5 ⊢ x ∈ V
22		oalim 3135	. . . . . 6 ⊢ ((∅ ∈ On ∧ (x ∈ V ∧ Lim x)) → (∅ +o x) = ∪y ∈ x (∅ +o y))
23	13, 22	mpan 518	. . . . 5 ⊢ ((x ∈ V ∧ Lim x) → (∅ +o x) = ∪y ∈ x (∅ +o y))
24	21, 23	mpan 518	. . . 4 ⊢ (Lim x → (∅ +o x) = ∪y ∈ x (∅ +o y))
25		limuni 2284	. . . 4 ⊢ (Lim x → x = ∪x)
26	24, 25	cleq12d 1115	. . 3 ⊢ (Lim x → ((∅ +o x) = x ↔ ∪y ∈ x (∅ +o y) = ∪x))
27		iuneq2 2006	. . . 4 ⊢ (∀y ∈ x (∅ +o y) = y → ∪y ∈ x (∅ +o y) = ∪y ∈ x y)
28		uniiun 2026	. . . 4 ⊢ ∪x = ∪y ∈ x y
29	27, 28	syl6eqr 1142	. . 3 ⊢ (∀y ∈ x (∅ +o y) = y → ∪y ∈ x (∅ +o y) = ∪x)
30	26, 29	syl5bir 184	. 2 ⊢ (Lim x → (∀y ∈ x (∅ +o y) = y → (∅ +o x) = x))
31	3, 6, 9, 12, 15, 20, 30	tfinds 2401	1 ⊢ (A ∈ On → (∅ +o A) = A)

Syntax hints:   → wi 2   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348  ∅c0 1707  ∪cuni 1919  ∪ciun 1994  Oncon0 2199  Lim wlim 2200  suc csuc 2201  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  om1 3144  oaword2 3155  nna0r 3170

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-opr 3003  df-oprab 3004  df-oadd 3106

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