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Theorem oa0r 8534
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 7410 . . 3 (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2740 . 2 (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4 oveq2 7410 . . 3 (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2740 . 2 (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7 oveq2 7410 . . 3 (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2740 . 2 (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10 oveq2 7410 . . 3 (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2740 . 2 (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13 0elon 6409 . . 3 ∅ ∈ On
14 oa0 8512 . . 3 (∅ ∈ On → (∅ +o ∅) = ∅)
1513, 14ax-mp 5 . 2 (∅ +o ∅) = ∅
16 oasuc 8520 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
1713, 16mpan 687 . . . 4 (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
18 suceq 6421 . . . 4 ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
1917, 18sylan9eq 2784 . . 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 5007 . . . 4 (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 𝑦𝑥 (∅ +o 𝑦) = 𝑦𝑥 𝑦)
22 uniiun 5052 . . . 4 𝑥 = 𝑦𝑥 𝑦
2321, 22eqtr4di 2782 . . 3 (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 𝑦𝑥 (∅ +o 𝑦) = 𝑥)
24 vex 3470 . . . . 5 𝑥 ∈ V
25 oalim 8528 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
2613, 25mpan 687 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
2724, 26mpan 687 . . . 4 (Lim 𝑥 → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
28 limuni 6416 . . . 4 (Lim 𝑥𝑥 = 𝑥)
2927, 28eqeq12d 2740 . . 3 (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 𝑦𝑥 (∅ +o 𝑦) = 𝑥))
3023, 29imbitrrid 245 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥))
313, 6, 9, 12, 15, 20, 30tfinds 7843 1 (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3053  Vcvv 3466  c0 4315   cuni 4900   ciun 4988  Oncon0 6355  Lim wlim 6356  suc csuc 6357  (class class class)co 7402   +o coa 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-oadd 8466
This theorem is referenced by:  om1  8538  oaword2  8549  oeeui  8598  oaabs2  8645  cantnfp1  9673  oaordnrex  42559  oacl2g  42594  tfsconcat0i  42609  ofoaf  42619  ofoaid2  42623
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