Theorem oasuc 6727

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 4643	. . . . . 6 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
2		oav2 6726	. . . . . 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 4511	. . . . . . . . . 10 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
5		iuneq1 4020	. . . . . . . . . 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 4087	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
8	6, 7	eqtri 2259	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
9		oveq2 6083	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
10		suceq 4542	. . . . . . . . . . 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 4083	. . . . . . . . 9 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
13	12	uneq2d 3383	. . . . . . . 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 2283	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
15	14	uneq2d 3383	. . . . . 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 2271	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
18		unass 3386	. . . 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 2289	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
20		oav2 6726	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)))
21	20	uneq1d 3382	. . 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 2274	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)))
23		sssucid 4555	. . 3 ⊢ (𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵)
24		ssequn1 3399	. . 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 2287	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   ∪ cun 3218   ⊆ wss 3220  {csn 3705  ∪ ciun 4007  Oncon0 4503  suc csuc 4505  (class class class)co 6075   +_o coa 6674

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681

This theorem is referenced by:  onasuc  6729  nnaordi  6771