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Theorem oa0r 7965
Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (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 6984 . . 3 (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2793 . 2 (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4 oveq2 6984 . . 3 (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2793 . 2 (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7 oveq2 6984 . . 3 (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2793 . 2 (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10 oveq2 6984 . . 3 (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2793 . 2 (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13 0elon 6082 . . 3 ∅ ∈ On
14 oa0 7943 . . 3 (∅ ∈ On → (∅ +o ∅) = ∅)
1513, 14ax-mp 5 . 2 (∅ +o ∅) = ∅
16 oasuc 7951 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
1713, 16mpan 677 . . . 4 (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
18 suceq 6094 . . . 4 ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
1917, 18sylan9eq 2834 . . 3 ((𝑦 ∈ On ∧ (∅ +o 𝑦) = 𝑦) → (∅ +o suc 𝑦) = suc 𝑦)
2019ex 405 . 2 (𝑦 ∈ On → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦))
21 iuneq2 4810 . . . 4 (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 𝑦𝑥 (∅ +o 𝑦) = 𝑦𝑥 𝑦)
22 uniiun 4848 . . . 4 𝑥 = 𝑦𝑥 𝑦
2321, 22syl6eqr 2832 . . 3 (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 𝑦𝑥 (∅ +o 𝑦) = 𝑥)
24 vex 3418 . . . . 5 𝑥 ∈ V
25 oalim 7959 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
2613, 25mpan 677 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
2724, 26mpan 677 . . . 4 (Lim 𝑥 → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
28 limuni 6089 . . . 4 (Lim 𝑥𝑥 = 𝑥)
2927, 28eqeq12d 2793 . . 3 (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 𝑦𝑥 (∅ +o 𝑦) = 𝑥))
3023, 29syl5ibr 238 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥))
313, 6, 9, 12, 15, 20, 30tfinds 7390 1 (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  wral 3088  Vcvv 3415  c0 4178   cuni 4712   ciun 4792  Oncon0 6029  Lim wlim 6030  suc csuc 6031  (class class class)co 6976   +o coa 7902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-oadd 7909
This theorem is referenced by:  om1  7969  oaword2  7980  oeeui  8029  oaabs2  8072  cantnfp1  8938
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