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Theorem oa0r 8576
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.
StepHypRef Expression
1 oveq2 7439 . . 3 (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2753 . 2 (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4 oveq2 7439 . . 3 (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2753 . 2 (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7 oveq2 7439 . . 3 (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2753 . 2 (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10 oveq2 7439 . . 3 (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2753 . 2 (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13 0elon 6438 . . 3 ∅ ∈ On
14 oa0 8554 . . 3 (∅ ∈ On → (∅ +o ∅) = ∅)
1513, 14ax-mp 5 . 2 (∅ +o ∅) = ∅
16 oasuc 8562 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
1713, 16mpan 690 . . . 4 (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
18 suceq 6450 . . . 4 ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
1917, 18sylan9eq 2797 . . 3 ((𝑦 ∈ On ∧ (∅ +o 𝑦) = 𝑦) → (∅ +o suc 𝑦) = suc 𝑦)
2019ex 412 . 2 (𝑦 ∈ On → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦))
21 iuneq2 5011 . . . 4 (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 𝑦𝑥 (∅ +o 𝑦) = 𝑦𝑥 𝑦)
22 uniiun 5058 . . . 4 𝑥 = 𝑦𝑥 𝑦
2321, 22eqtr4di 2795 . . 3 (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 𝑦𝑥 (∅ +o 𝑦) = 𝑥)
24 vex 3484 . . . . 5 𝑥 ∈ V
25 oalim 8570 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
2613, 25mpan 690 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
2724, 26mpan 690 . . . 4 (Lim 𝑥 → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
28 limuni 6445 . . . 4 (Lim 𝑥𝑥 = 𝑥)
2927, 28eqeq12d 2753 . . 3 (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 𝑦𝑥 (∅ +o 𝑦) = 𝑥))
3023, 29imbitrrid 246 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥))
313, 6, 9, 12, 15, 20, 30tfinds 7881 1 (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  c0 4333   cuni 4907   ciun 4991  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431   +o coa 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510
This theorem is referenced by:  om1  8580  oaword2  8591  oeeui  8640  oaabs2  8687  cantnfp1  9721  oaordnrex  43308  oacl2g  43343  tfsconcat0i  43358  ofoaf  43368  ofoaid2  43372
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