Theorem oa0r 8495

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 7393	. . 3 ⊢ (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2772	. 2 ⊢ (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4		oveq2 7393	. . 3 ⊢ (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2772	. 2 ⊢ (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7		oveq2 7393	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8		id 22	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
9	7, 8	eqeq12d 2772	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10		oveq2 7393	. . 3 ⊢ (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2772	. 2 ⊢ (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13		0elon 6390	. . 3 ⊢ ∅ ∈ On
14		oa0 8473	. . 3 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (∅ +o ∅) = ∅
16		oasuc 8481	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
17	13, 16	mpan 698	. . . 4 ⊢ (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
18		suceq 6403	. . . 4 ⊢ ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
19	17, 18	sylan9eq 2811	. . 3 ⊢ ((𝑦 ∈ On ∧ (∅ +o 𝑦) = 𝑦) → (∅ +o suc 𝑦) = suc 𝑦)
20	19	ex 415	. 2 ⊢ (𝑦 ∈ On → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦))
21		iuneq2 4963	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
22		uniiun 5010	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
23	21, 22	eqtr4di 2809	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥)
24		vex 3452	. . . . 5 ⊢ 𝑥 ∈ V
25		oalim 8489	. . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
26	13, 25	mpan 698	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
27	24, 26	mpan 698	. . . 4 ⊢ (Lim 𝑥 → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
28		limuni 6397	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
29	27, 28	eqeq12d 2772	. . 3 ⊢ (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥))
30	23, 29	imbitrrid 248	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥))
31	3, 6, 9, 12, 15, 20, 30	tfinds 7829	1 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∀wral 3070  Vcvv 3448  ∅c0 4280  ∪ cuni 4859  ∪ ciun 4943  Oncon0 6335  Lim wlim 6336  suc csuc 6337  (class class class)co 7385   +_o coa 8422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-oadd 8429

This theorem is referenced by:  om1  8499  oaword2  8510  oeeui  8560  oaabs2  8607  cantnfp1  9626  fineqvnttrclse  35375  oaordnrex  43820  oacl2g  43855  tfsconcat0i  43870  ofoaf  43880  ofoaid2  43884