Theorem nna0r 7689

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 7618) so that we can avoid ax-rep 4771, 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.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . 3 ⊢ (𝑥 = ∅ → (∅ +𝑜 𝑥) = (∅ +𝑜 ∅))
2		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2637	. 2 ⊢ (𝑥 = ∅ → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 ∅) = ∅))
4		oveq2 6658	. . 3 ⊢ (𝑥 = 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2637	. 2 ⊢ (𝑥 = 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝑦) = 𝑦))
7		oveq2 6658	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 suc 𝑦))
8		id 22	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
9	7, 8	eqeq12d 2637	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦))
10		oveq2 6658	. . 3 ⊢ (𝑥 = 𝐴 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝐴))
11		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2637	. 2 ⊢ (𝑥 = 𝐴 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝐴) = 𝐴))
13		0elon 5778	. . 3 ⊢ ∅ ∈ On
14		oa0 7596	. . 3 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (∅ +𝑜 ∅) = ∅
16		peano1 7085	. . . 4 ⊢ ∅ ∈ ω
17		nnasuc 7686	. . . 4 ⊢ ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
18	16, 17	mpan 706	. . 3 ⊢ (𝑦 ∈ ω → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
19		suceq 5790	. . . 4 ⊢ ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦)
20	19	eqeq2d 2632	. . 3 ⊢ ((∅ +𝑜 𝑦) = 𝑦 → ((∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦) ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦))
21	18, 20	syl5ibcom 235	. 2 ⊢ (𝑦 ∈ ω → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦))
22	3, 6, 9, 12, 15, 21	finds 7092	1 ⊢ (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∅c0 3915  Oncon0 5723  suc csuc 5725  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  nnacom  7697  nnm1  7728