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Theorem oa1suc 7498
 Description: Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
oa1suc (𝐴 ∈ On → (𝐴 +𝑜 1𝑜) = suc 𝐴)

Proof of Theorem oa1suc
StepHypRef Expression
1 df-1o 7447 . . . 4 1𝑜 = suc ∅
21oveq2i 6560 . . 3 (𝐴 +𝑜 1𝑜) = (𝐴 +𝑜 suc ∅)
3 peano1 6977 . . . 4 ∅ ∈ ω
4 onasuc 7495 . . . 4 ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 +𝑜 suc ∅) = suc (𝐴 +𝑜 ∅))
53, 4mpan2 703 . . 3 (𝐴 ∈ On → (𝐴 +𝑜 suc ∅) = suc (𝐴 +𝑜 ∅))
62, 5syl5eq 2656 . 2 (𝐴 ∈ On → (𝐴 +𝑜 1𝑜) = suc (𝐴 +𝑜 ∅))
7 oa0 7483 . . 3 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
8 suceq 5707 . . 3 ((𝐴 +𝑜 ∅) = 𝐴 → suc (𝐴 +𝑜 ∅) = suc 𝐴)
97, 8syl 17 . 2 (𝐴 ∈ On → suc (𝐴 +𝑜 ∅) = suc 𝐴)
106, 9eqtrd 2644 1 (𝐴 ∈ On → (𝐴 +𝑜 1𝑜) = suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∅c0 3874  Oncon0 5640  suc csuc 5642  (class class class)co 6549  ωcom 6957  1𝑜c1o 7440   +𝑜 coa 7444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451 This theorem is referenced by:  o1p1e2  7507  o2p2e4  7508  om1r  7510  omlimcl  7545  oneo  7548  oeeui  7569  nnneo  7618  nneob  7619  oancom  8431  indpi  9608
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