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Theorem nna0r 6304
Description: Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Assertion
Ref Expression
nna0r (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)

Proof of Theorem nna0r
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5714 . . 3 (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2 id 19 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2114 . 2 (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4 oveq2 5714 . . 3 (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5 id 19 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2114 . 2 (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7 oveq2 5714 . . 3 (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8 id 19 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2114 . 2 (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10 oveq2 5714 . . 3 (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11 id 19 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2114 . 2 (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13 0elon 4252 . . 3 ∅ ∈ On
14 oa0 6283 . . 3 (∅ ∈ On → (∅ +o ∅) = ∅)
1513, 14ax-mp 7 . 2 (∅ +o ∅) = ∅
16 peano1 4446 . . . 4 ∅ ∈ ω
17 nnasuc 6302 . . . 4 ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
1816, 17mpan 418 . . 3 (𝑦 ∈ ω → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
19 suceq 4262 . . . 4 ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
2019eqeq2d 2111 . . 3 ((∅ +o 𝑦) = 𝑦 → ((∅ +o suc 𝑦) = suc (∅ +o 𝑦) ↔ (∅ +o suc 𝑦) = suc 𝑦))
2118, 20syl5ibcom 154 . 2 (𝑦 ∈ ω → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦))
223, 6, 9, 12, 15, 21finds 4452 1 (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wcel 1448  c0 3310  Oncon0 4223  suc csuc 4225  ωcom 4442  (class class class)co 5706   +o coa 6240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-oadd 6247
This theorem is referenced by:  nnacom  6310  nnaword  6337  nnm1  6350  prarloclem5  7209
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