Theorem oa0r 8511

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 7408	. . 3 ⊢ (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2		id 23	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
3	1, 2	eqeq12d 2781	. 2 ⊢ (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4		oveq2 7408	. . 3 ⊢ (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5		id 23	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6	4, 5	eqeq12d 2781	. 2 ⊢ (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7		oveq2 7408	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8		id 23	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
9	7, 8	eqeq12d 2781	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10		oveq2 7408	. . 3 ⊢ (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11		id 23	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12	10, 11	eqeq12d 2781	. 2 ⊢ (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13		0elon 6405	. . 3 ⊢ ∅ ∈ On
14		oa0 8489	. . 3 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (∅ +o ∅) = ∅
16		oasuc 8497	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
17	13, 16	mpan 702	. . . 4 ⊢ (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
18		suceq 6418	. . . 4 ⊢ ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
19	17, 18	sylan9eq 2820	. . 3 ⊢ ((𝑦 ∈ On ∧ (∅ +o 𝑦) = 𝑦) → (∅ +o suc 𝑦) = suc 𝑦)
20	19	ex 417	. 2 ⊢ (𝑦 ∈ On → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦))
21		iuneq2 4972	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑦 ∈ 𝑥 𝑦)
22		uniiun 5019	. . . 4 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
23	21, 22	eqtr4di 2818	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥)
24		vex 3461	. . . . 5 ⊢ 𝑥 ∈ V
25		oalim 8505	. . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
26	13, 25	mpan 702	. . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
27	24, 26	mpan 702	. . . 4 ⊢ (Lim 𝑥 → (∅ +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦))
28		limuni 6412	. . . 4 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
29	27, 28	eqeq12d 2781	. . 3 ⊢ (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 ↔ ∪ 𝑦 ∈ 𝑥 (∅ +o 𝑦) = ∪ 𝑥))
30	23, 29	imbitrrid 249	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥))
31	3, 6, 9, 12, 15, 20, 30	tfinds 7844	1 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457  ∅c0 4288  ∪ cuni 4868  ∪ ciun 4952  Oncon0 6350  Lim wlim 6351  suc csuc 6352  (class class class)co 7400   +_o coa 8438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-oadd 8445

This theorem is referenced by:  om1  8515  oaword2  8526  oeeui  8576  oaabs2  8623  cantnfp1  9638  fineqvnttrclse  35432  oaordnrex  43884  oacl2g  43919  tfsconcat0i  43934  ofoaf  43944  ofoaid2  43948