ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nna0r Structured version   GIF version

Theorem nna0r 5968
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 5440 . . 3 (x = ∅ → (∅ +𝑜 x) = (∅ +𝑜 ∅))
2 id 19 . . 3 (x = ∅ → x = ∅)
31, 2eqeq12d 2032 . 2 (x = ∅ → ((∅ +𝑜 x) = x ↔ (∅ +𝑜 ∅) = ∅))
4 oveq2 5440 . . 3 (x = y → (∅ +𝑜 x) = (∅ +𝑜 y))
5 id 19 . . 3 (x = yx = y)
64, 5eqeq12d 2032 . 2 (x = y → ((∅ +𝑜 x) = x ↔ (∅ +𝑜 y) = y))
7 oveq2 5440 . . 3 (x = suc y → (∅ +𝑜 x) = (∅ +𝑜 suc y))
8 id 19 . . 3 (x = suc yx = suc y)
97, 8eqeq12d 2032 . 2 (x = suc y → ((∅ +𝑜 x) = x ↔ (∅ +𝑜 suc y) = suc y))
10 oveq2 5440 . . 3 (x = A → (∅ +𝑜 x) = (∅ +𝑜 A))
11 id 19 . . 3 (x = Ax = A)
1210, 11eqeq12d 2032 . 2 (x = A → ((∅ +𝑜 x) = x ↔ (∅ +𝑜 A) = A))
13 0elon 4074 . . 3 On
14 oa0 5948 . . 3 (∅ On → (∅ +𝑜 ∅) = ∅)
1513, 14ax-mp 7 . 2 (∅ +𝑜 ∅) = ∅
16 peano1 4240 . . . 4 𝜔
17 nnasuc 5966 . . . 4 ((∅ 𝜔 y 𝜔) → (∅ +𝑜 suc y) = suc (∅ +𝑜 y))
1816, 17mpan 402 . . 3 (y 𝜔 → (∅ +𝑜 suc y) = suc (∅ +𝑜 y))
19 suceq 4084 . . . 4 ((∅ +𝑜 y) = y → suc (∅ +𝑜 y) = suc y)
2019eqeq2d 2029 . . 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 4246 1 (A 𝜔 → (∅ +𝑜 A) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1226   wcel 1370  c0 3197  Oncon0 4045  suc csuc 4047  𝜔com 4236  (class class class)co 5432   +𝑜 coa 5909
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-oadd 5916
This theorem is referenced by:  nnacom  5974  nnaword  5991  nnm1  6004  prarloclem5  6348
  Copyright terms: Public domain W3C validator