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Theorem nna0r 5996
 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 5463 . . 3 (x = ∅ → (∅ +𝑜 x) = (∅ +𝑜 ∅))
2 id 19 . . 3 (x = ∅ → x = ∅)
31, 2eqeq12d 2051 . 2 (x = ∅ → ((∅ +𝑜 x) = x ↔ (∅ +𝑜 ∅) = ∅))
4 oveq2 5463 . . 3 (x = y → (∅ +𝑜 x) = (∅ +𝑜 y))
5 id 19 . . 3 (x = yx = y)
64, 5eqeq12d 2051 . 2 (x = y → ((∅ +𝑜 x) = x ↔ (∅ +𝑜 y) = y))
7 oveq2 5463 . . 3 (x = suc y → (∅ +𝑜 x) = (∅ +𝑜 suc y))
8 id 19 . . 3 (x = suc yx = suc y)
97, 8eqeq12d 2051 . 2 (x = suc y → ((∅ +𝑜 x) = x ↔ (∅ +𝑜 suc y) = suc y))
10 oveq2 5463 . . 3 (x = A → (∅ +𝑜 x) = (∅ +𝑜 A))
11 id 19 . . 3 (x = Ax = A)
1210, 11eqeq12d 2051 . 2 (x = A → ((∅ +𝑜 x) = x ↔ (∅ +𝑜 A) = A))
13 0elon 4095 . . 3 On
14 oa0 5976 . . 3 (∅ On → (∅ +𝑜 ∅) = ∅)
1513, 14ax-mp 7 . 2 (∅ +𝑜 ∅) = ∅
16 peano1 4260 . . . 4 𝜔
17 nnasuc 5994 . . . 4 ((∅ 𝜔 y 𝜔) → (∅ +𝑜 suc y) = suc (∅ +𝑜 y))
1816, 17mpan 400 . . 3 (y 𝜔 → (∅ +𝑜 suc y) = suc (∅ +𝑜 y))
19 suceq 4105 . . . 4 ((∅ +𝑜 y) = y → suc (∅ +𝑜 y) = suc y)
2019eqeq2d 2048 . . 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 4266 1 (A 𝜔 → (∅ +𝑜 A) = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ∅c0 3218  Oncon0 4066  suc csuc 4068  𝜔com 4256  (class class class)co 5455   +𝑜 coa 5937 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944 This theorem is referenced by:  nnacom  6002  nnaword  6020  nnm1  6033  prarloclem5  6482
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