Theorem oasuc 6631

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 4599	. . . . . 6 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
2		oav2 6630	. . . . . 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 4468	. . . . . . . . . 10 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
5		iuneq1 3983	. . . . . . . . . 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 4050	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
8	6, 7	eqtri 2252	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
9		oveq2 6025	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
10		suceq 4499	. . . . . . . . . . 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 4046	. . . . . . . . 9 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
13	12	uneq2d 3361	. . . . . . . 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 2276	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
15	14	uneq2d 3361	. . . . . 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 2264	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
18		unass 3364	. . . 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 2282	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
20		oav2 6630	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)))
21	20	uneq1d 3360	. . 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 2267	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)))
23		sssucid 4512	. . 3 ⊢ (𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵)
24		ssequn1 3377	. . 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 2280	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202   ∪ cun 3198   ⊆ wss 3200  {csn 3669  ∪ ciun 3970  Oncon0 4460  suc csuc 4462  (class class class)co 6017   +_o coa 6578

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-oadd 6585

This theorem is referenced by:  onasuc  6633  nnaordi  6675