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Theorem nna0r 7674
 Description: Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. Note: In this and later theorems, we deliberately avoid the more general ordinal versions of these theorems (in this case oa0r 7603) so that we can avoid ax-rep 4762, which is not needed for finite recursive definitions. (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 6643 . . 3 (𝑥 = ∅ → (∅ +𝑜 𝑥) = (∅ +𝑜 ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2635 . 2 (𝑥 = ∅ → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 ∅) = ∅))
4 oveq2 6643 . . 3 (𝑥 = 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2635 . 2 (𝑥 = 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝑦) = 𝑦))
7 oveq2 6643 . . 3 (𝑥 = suc 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2635 . 2 (𝑥 = suc 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦))
10 oveq2 6643 . . 3 (𝑥 = 𝐴 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2635 . 2 (𝑥 = 𝐴 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝐴) = 𝐴))
13 0elon 5766 . . 3 ∅ ∈ On
14 oa0 7581 . . 3 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
1513, 14ax-mp 5 . 2 (∅ +𝑜 ∅) = ∅
16 peano1 7070 . . . 4 ∅ ∈ ω
17 nnasuc 7671 . . . 4 ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
1816, 17mpan 705 . . 3 (𝑦 ∈ ω → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
19 suceq 5778 . . . 4 ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦)
2019eqeq2d 2630 . . 3 ((∅ +𝑜 𝑦) = 𝑦 → ((∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦) ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦))
2118, 20syl5ibcom 235 . 2 (𝑦 ∈ ω → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦))
223, 6, 9, 12, 15, 21finds 7077 1 (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1481   ∈ wcel 1988  ∅c0 3907  Oncon0 5711  suc csuc 5713  (class class class)co 6635  ωcom 7050   +𝑜 coa 7542 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549 This theorem is referenced by:  nnacom  7682  nnm1  7713
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