Theorem oa0r 8473

Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. Lemma 2.14 of [Schloeder] p. 5. (Contributed by NM, 5-May-1995.)

Assertion

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

Proof of Theorem oa0r

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7375	. . 3 ⊢ (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2752	. 2 ⊢ (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4		oveq2 7375	. . 3 ⊢ (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7		oveq2 7375	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8		id 22	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
9	7, 8	eqeq12d 2752	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10		oveq2 7375	. . 3 ⊢ (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13		0elon 6378	. . 3 ⊢ ∅ ∈ On
14		oa0 8451	. . 3 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (∅ +o ∅) = ∅
16		oasuc 8459	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
17	13, 16	mpan 691	. . . 4 ⊢ (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
18		suceq 6391	. . . 4 ⊢ ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
19	17, 18	sylan9eq 2791	. . 3 ⊢ ((𝑦 ∈ On ∧ (∅ +o 𝑦) = 𝑦) → (∅ +o suc 𝑦) = suc 𝑦)
20	19	ex 412	. 2 ⊢ (𝑦 ∈ On → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦))
21		iuneq2 4953	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
22		uniiun 5001	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
23	21, 22	eqtr4di 2789	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥)
24		vex 3433	. . . . 5 ⊢ 𝑥 ∈ V
25		oalim 8467	. . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
26	13, 25	mpan 691	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
27	24, 26	mpan 691	. . . 4 ⊢ (Lim 𝑥 → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
28		limuni 6385	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
29	27, 28	eqeq12d 2752	. . 3 ⊢ (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥))
30	23, 29	imbitrrid 246	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥))
31	3, 6, 9, 12, 15, 20, 30	tfinds 7811	1 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429  ∅c0 4273  ∪ cuni 4850  ∪ ciun 4933  Oncon0 6323  Lim wlim 6324  suc csuc 6325  (class class class)co 7367   +_o coa 8402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409

This theorem is referenced by:  om1  8477  oaword2  8488  oeeui  8538  oaabs2  8585  cantnfp1  9602  fineqvnttrclse  35268  oaordnrex  43723  oacl2g  43758  tfsconcat0i  43773  ofoaf  43783  ofoaid2  43787