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Theorem oa0r 7505
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 6557 . . 3 (𝑥 = ∅ → (∅ +𝑜 𝑥) = (∅ +𝑜 ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2625 . 2 (𝑥 = ∅ → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 ∅) = ∅))
4 oveq2 6557 . . 3 (𝑥 = 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2625 . 2 (𝑥 = 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝑦) = 𝑦))
7 oveq2 6557 . . 3 (𝑥 = suc 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2625 . 2 (𝑥 = suc 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦))
10 oveq2 6557 . . 3 (𝑥 = 𝐴 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2625 . 2 (𝑥 = 𝐴 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝐴) = 𝐴))
13 0elon 5695 . . 3 ∅ ∈ On
14 oa0 7483 . . 3 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
1513, 14ax-mp 5 . 2 (∅ +𝑜 ∅) = ∅
16 oasuc 7491 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
1713, 16mpan 702 . . . 4 (𝑦 ∈ On → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
18 suceq 5707 . . . 4 ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦)
1917, 18sylan9eq 2664 . . 3 ((𝑦 ∈ On ∧ (∅ +𝑜 𝑦) = 𝑦) → (∅ +𝑜 suc 𝑦) = suc 𝑦)
2019ex 449 . 2 (𝑦 ∈ On → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦))
21 iuneq2 4473 . . . 4 (∀𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦 𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦𝑥 𝑦)
22 uniiun 4509 . . . 4 𝑥 = 𝑦𝑥 𝑦
2321, 22syl6eqr 2662 . . 3 (∀𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦 𝑦𝑥 (∅ +𝑜 𝑦) = 𝑥)
24 vex 3176 . . . . 5 𝑥 ∈ V
25 oalim 7499 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +𝑜 𝑥) = 𝑦𝑥 (∅ +𝑜 𝑦))
2613, 25mpan 702 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +𝑜 𝑥) = 𝑦𝑥 (∅ +𝑜 𝑦))
2724, 26mpan 702 . . . 4 (Lim 𝑥 → (∅ +𝑜 𝑥) = 𝑦𝑥 (∅ +𝑜 𝑦))
28 limuni 5702 . . . 4 (Lim 𝑥𝑥 = 𝑥)
2927, 28eqeq12d 2625 . . 3 (Lim 𝑥 → ((∅ +𝑜 𝑥) = 𝑥 𝑦𝑥 (∅ +𝑜 𝑦) = 𝑥))
3023, 29syl5ibr 235 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 𝑥) = 𝑥))
313, 6, 9, 12, 15, 20, 30tfinds 6951 1 (𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  c0 3874   cuni 4372   ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +𝑜 coa 7444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451
This theorem is referenced by:  om1  7509  oaword2  7520  oeeui  7569  oaabs2  7612  cantnfp1  8461
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