Theorem oasuc 6697

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		onsuc 4623	. . . . . 6 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
2		oav2 6696	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)))
3	1, 2	sylan2 286	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)))
4		df-suc 4492	. . . . . . . . . 10 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
5		iuneq1 4004	. . . . . . . . . 10 ⊢ (suc 𝐵 = (𝐵 ∪ {𝐵}) → ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥))
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥)
7		iunxun 4071	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
8	6, 7	eqtri 2253	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
9		oveq2 6058	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
10		suceq 4523	. . . . . . . . . . 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 4067	. . . . . . . . 9 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
13	12	uneq2d 3373	. . . . . . . 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	eqtrid 2277	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
15	14	uneq2d 3373	. . . . . 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 277	. . . . 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 2265	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
18		unass 3376	. . . 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	eqtr4di 2283	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
20		oav2 6696	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)))
21	20	uneq1d 3372	. . 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 2268	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)))
23		sssucid 4536	. . 3 ⊢ (𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵)
24		ssequn1 3389	. . 3 ⊢ ((𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵) ↔ ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵))
25	23, 24	mpbi 145	. 2 ⊢ ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵)
26	22, 25	eqtrdi 2281	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203   ∪ cun 3209   ⊆ wss 3211  {csn 3689  ∪ ciun 3991  Oncon0 4484  suc csuc 4486  (class class class)co 6050   +_o coa 6644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-oadd 6651

This theorem is referenced by:  onasuc  6699  nnaordi  6741