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Theorem oa0r 7773
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 6802 . . 3 (𝑥 = ∅ → (∅ +𝑜 𝑥) = (∅ +𝑜 ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2786 . 2 (𝑥 = ∅ → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 ∅) = ∅))
4 oveq2 6802 . . 3 (𝑥 = 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2786 . 2 (𝑥 = 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝑦) = 𝑦))
7 oveq2 6802 . . 3 (𝑥 = suc 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2786 . 2 (𝑥 = suc 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦))
10 oveq2 6802 . . 3 (𝑥 = 𝐴 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2786 . 2 (𝑥 = 𝐴 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝐴) = 𝐴))
13 0elon 5922 . . 3 ∅ ∈ On
14 oa0 7751 . . 3 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
1513, 14ax-mp 5 . 2 (∅ +𝑜 ∅) = ∅
16 oasuc 7759 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
1713, 16mpan 664 . . . 4 (𝑦 ∈ On → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
18 suceq 5934 . . . 4 ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦)
1917, 18sylan9eq 2825 . . 3 ((𝑦 ∈ On ∧ (∅ +𝑜 𝑦) = 𝑦) → (∅ +𝑜 suc 𝑦) = suc 𝑦)
2019ex 397 . 2 (𝑦 ∈ On → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦))
21 iuneq2 4672 . . . 4 (∀𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦 𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦𝑥 𝑦)
22 uniiun 4708 . . . 4 𝑥 = 𝑦𝑥 𝑦
2321, 22syl6eqr 2823 . . 3 (∀𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦 𝑦𝑥 (∅ +𝑜 𝑦) = 𝑥)
24 vex 3354 . . . . 5 𝑥 ∈ V
25 oalim 7767 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +𝑜 𝑥) = 𝑦𝑥 (∅ +𝑜 𝑦))
2613, 25mpan 664 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +𝑜 𝑥) = 𝑦𝑥 (∅ +𝑜 𝑦))
2724, 26mpan 664 . . . 4 (Lim 𝑥 → (∅ +𝑜 𝑥) = 𝑦𝑥 (∅ +𝑜 𝑦))
28 limuni 5929 . . . 4 (Lim 𝑥𝑥 = 𝑥)
2927, 28eqeq12d 2786 . . 3 (Lim 𝑥 → ((∅ +𝑜 𝑥) = 𝑥 𝑦𝑥 (∅ +𝑜 𝑦) = 𝑥))
3023, 29syl5ibr 236 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 𝑥) = 𝑥))
313, 6, 9, 12, 15, 20, 30tfinds 7207 1 (𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  c0 4064   cuni 4575   ciun 4655  Oncon0 5867  Lim wlim 5868  suc csuc 5869  (class class class)co 6794   +𝑜 coa 7711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-oadd 7718
This theorem is referenced by:  om1  7777  oaword2  7788  oeeui  7837  oaabs2  7880  cantnfp1  8743
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