Theorem oa0r 7618

Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (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 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		oasuc 7604	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
17	13, 16	mpan 706	. . . 4 ⊢ (𝑦 ∈ On → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
18		suceq 5790	. . . 4 ⊢ ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦)
19	17, 18	sylan9eq 2676	. . 3 ⊢ ((𝑦 ∈ On ∧ (∅ +𝑜 𝑦) = 𝑦) → (∅ +𝑜 suc 𝑦) = suc 𝑦)
20	19	ex 450	. 2 ⊢ (𝑦 ∈ On → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦))
21		iuneq2 4537	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
22		uniiun 4573	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
23	21, 22	syl6eqr 2674	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = ∪ 𝑥)
24		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
25		oalim 7612	. . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦))
26	13, 25	mpan 706	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦))
27	24, 26	mpan 706	. . . 4 ⊢ (Lim 𝑥 → (∅ +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦))
28		limuni 5785	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
29	27, 28	eqeq12d 2637	. . 3 ⊢ (Lim 𝑥 → ((∅ +𝑜 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = ∪ 𝑥))
30	23, 29	syl5ibr 236	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 𝑥) = 𝑥))
31	3, 6, 9, 12, 15, 20, 30	tfinds 7059	1 ⊢ (𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  ∅c0 3915  ∪ cuni 4436  ∪ ciun 4520  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650   +_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-rep 4771  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:  om1  7622  oaword2  7633  oeeui  7682  oaabs2  7725  cantnfp1  8578