Theorem nna0r 6536

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.

Step	Hyp	Ref	Expression
1		oveq2 5930	. . 3 ⊢ (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2		id 19	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2211	. 2 ⊢ (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4		oveq2 5930	. . 3 ⊢ (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5		id 19	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2211	. 2 ⊢ (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7		oveq2 5930	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8		id 19	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
9	7, 8	eqeq12d 2211	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10		oveq2 5930	. . 3 ⊢ (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11		id 19	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2211	. 2 ⊢ (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13		0elon 4427	. . 3 ⊢ ∅ ∈ On
14		oa0 6515	. . 3 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (∅ +o ∅) = ∅
16		peano1 4630	. . . 4 ⊢ ∅ ∈ ω
17		nnasuc 6534	. . . 4 ⊢ ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
18	16, 17	mpan 424	. . 3 ⊢ (𝑦 ∈ ω → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
19		suceq 4437	. . . 4 ⊢ ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
20	19	eqeq2d 2208	. . 3 ⊢ ((∅ +o 𝑦) = 𝑦 → ((∅ +o suc 𝑦) = suc (∅ +o 𝑦) ↔ (∅ +o suc 𝑦) = suc 𝑦))
21	18, 20	syl5ibcom 155	. 2 ⊢ (𝑦 ∈ ω → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦))
22	3, 6, 9, 12, 15, 21	finds 4636	1 ⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  ∅c0 3450  Oncon0 4398  suc csuc 4400  ωcom 4626  (class class class)co 5922   +_o coa 6471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-oadd 6478

This theorem is referenced by:  nnacom  6542  nnaword  6569  nnm1  6583  prarloclem5  7567