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Theorem nna0r 6057
 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 5520 . . 3 (𝑥 = ∅ → (∅ +𝑜 𝑥) = (∅ +𝑜 ∅))
2 id 19 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2054 . 2 (𝑥 = ∅ → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 ∅) = ∅))
4 oveq2 5520 . . 3 (𝑥 = 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝑦))
5 id 19 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2054 . 2 (𝑥 = 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝑦) = 𝑦))
7 oveq2 5520 . . 3 (𝑥 = suc 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 suc 𝑦))
8 id 19 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2054 . 2 (𝑥 = suc 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦))
10 oveq2 5520 . . 3 (𝑥 = 𝐴 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝐴))
11 id 19 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2054 . 2 (𝑥 = 𝐴 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝐴) = 𝐴))
13 0elon 4129 . . 3 ∅ ∈ On
14 oa0 6037 . . 3 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
1513, 14ax-mp 7 . 2 (∅ +𝑜 ∅) = ∅
16 peano1 4317 . . . 4 ∅ ∈ ω
17 nnasuc 6055 . . . 4 ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
1816, 17mpan 400 . . 3 (𝑦 ∈ ω → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
19 suceq 4139 . . . 4 ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦)
2019eqeq2d 2051 . . 3 ((∅ +𝑜 𝑦) = 𝑦 → ((∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦) ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦))
2118, 20syl5ibcom 144 . 2 (𝑦 ∈ ω → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦))
223, 6, 9, 12, 15, 21finds 4323 1 (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ∅c0 3224  Oncon0 4100  suc csuc 4102  ωcom 4313  (class class class)co 5512   +𝑜 coa 5998 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005 This theorem is referenced by:  nnacom  6063  nnaword  6084  nnm1  6097  prarloclem5  6598
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