Theorem oa0r 8537

Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. Lemma 2.14 of [Schloeder] p. 5. (Contributed by NM, 5-May-1995.)

Assertion

Ref	Expression
oa0r	⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)

Proof of Theorem oa0r

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7416	. . 3 ⊢ (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2748	. 2 ⊢ (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4		oveq2 7416	. . 3 ⊢ (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2748	. 2 ⊢ (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7		oveq2 7416	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8		id 22	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
9	7, 8	eqeq12d 2748	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10		oveq2 7416	. . 3 ⊢ (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2748	. 2 ⊢ (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13		0elon 6418	. . 3 ⊢ ∅ ∈ On
14		oa0 8515	. . 3 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (∅ +o ∅) = ∅
16		oasuc 8523	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
17	13, 16	mpan 688	. . . 4 ⊢ (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
18		suceq 6430	. . . 4 ⊢ ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
19	17, 18	sylan9eq 2792	. . 3 ⊢ ((𝑦 ∈ On ∧ (∅ +o 𝑦) = 𝑦) → (∅ +o suc 𝑦) = suc 𝑦)
20	19	ex 413	. 2 ⊢ (𝑦 ∈ On → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦))
21		iuneq2 5016	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
22		uniiun 5061	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
23	21, 22	eqtr4di 2790	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥)
24		vex 3478	. . . . 5 ⊢ 𝑥 ∈ V
25		oalim 8531	. . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
26	13, 25	mpan 688	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
27	24, 26	mpan 688	. . . 4 ⊢ (Lim 𝑥 → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
28		limuni 6425	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
29	27, 28	eqeq12d 2748	. . 3 ⊢ (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥))
30	23, 29	imbitrrid 245	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥))
31	3, 6, 9, 12, 15, 20, 30	tfinds 7848	1 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474  ∅c0 4322  ∪ cuni 4908  ∪ ciun 4997  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7408   +_o coa 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-oadd 8469

This theorem is referenced by:  om1  8541  oaword2  8552  oeeui  8601  oaabs2  8647  cantnfp1  9675  oaordnrex  42035  oacl2g  42070  tfsconcat0i  42085  ofoaf  42095  ofoaid2  42099