Theorem oasuc 6067

Description: Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oasuc	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Proof of Theorem oasuc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suceloni 4245	. . . . . 6 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
2		oav2 6066	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥)))
3	1, 2	sylan2 280	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥)))
4		df-suc 4126	. . . . . . . . . 10 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
5		iuneq1 3691	. . . . . . . . . 10 ⊢ (suc 𝐵 = (𝐵 ∪ {𝐵}) → ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +𝑜 𝑥))
6	4, 5	ax-mp 7	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +𝑜 𝑥)
7		iunxun 3756	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +𝑜 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +𝑜 𝑥))
8	6, 7	eqtri 2101	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +𝑜 𝑥))
9		oveq2 5540	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵))
10		suceq 4157	. . . . . . . . . . 11 ⊢ ((𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵) → suc (𝐴 +𝑜 𝑥) = suc (𝐴 +𝑜 𝐵))
11	9, 10	syl 14	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → suc (𝐴 +𝑜 𝑥) = suc (𝐴 +𝑜 𝐵))
12	11	iunxsng 3753	. . . . . . . . 9 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵}suc (𝐴 +𝑜 𝑥) = suc (𝐴 +𝑜 𝐵))
13	12	uneq2d 3126	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +𝑜 𝑥)) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ suc (𝐴 +𝑜 𝐵)))
14	8, 13	syl5eq 2125	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ suc (𝐴 +𝑜 𝐵)))
15	14	uneq2d 3126	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥)) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ suc (𝐴 +𝑜 𝐵))))
16	15	adantl 271	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥)) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ suc (𝐴 +𝑜 𝐵))))
17	3, 16	eqtrd 2113	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ suc (𝐴 +𝑜 𝐵))))
18		unass 3129	. . . 4 ⊢ ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥)) ∪ suc (𝐴 +𝑜 𝐵)) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ suc (𝐴 +𝑜 𝐵)))
19	17, 18	syl6eqr 2131	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥)) ∪ suc (𝐴 +𝑜 𝐵)))
20		oav2 6066	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥)))
21	20	uneq1d 3125	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) ∪ suc (𝐴 +𝑜 𝐵)) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥)) ∪ suc (𝐴 +𝑜 𝐵)))
22	19, 21	eqtr4d 2116	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = ((𝐴 +𝑜 𝐵) ∪ suc (𝐴 +𝑜 𝐵)))
23		sssucid 4170	. . 3 ⊢ (𝐴 +𝑜 𝐵) ⊆ suc (𝐴 +𝑜 𝐵)
24		ssequn1 3142	. . 3 ⊢ ((𝐴 +𝑜 𝐵) ⊆ suc (𝐴 +𝑜 𝐵) ↔ ((𝐴 +𝑜 𝐵) ∪ suc (𝐴 +𝑜 𝐵)) = suc (𝐴 +𝑜 𝐵))
25	23, 24	mpbi 143	. 2 ⊢ ((𝐴 +𝑜 𝐵) ∪ suc (𝐴 +𝑜 𝐵)) = suc (𝐴 +𝑜 𝐵)
26	22, 25	syl6eq 2129	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433   ∪ cun 2971   ⊆ wss 2973  {csn 3398  ∪ ciun 3678  Oncon0 4118  suc csuc 4120  (class class class)co 5532   +_𝑜 coa 6021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028

This theorem is referenced by:  onasuc  6069  nnaordi  6104