Theorem oasuc 6044

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 4227	. . . . . 6 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
2		oav2 6043	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥)))
3	1, 2	sylan2 270	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥)))
4		df-suc 4108	. . . . . . . . . 10 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
5		iuneq1 3670	. . . . . . . . . 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 3735	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +𝑜 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +𝑜 𝑥))
8	6, 7	eqtri 2060	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ ∪ 𝑥 ∈ {𝐵}suc (𝐴 +𝑜 𝑥))
9		oveq2 5520	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵))
10		suceq 4139	. . . . . . . . . . 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 3732	. . . . . . . . 9 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵}suc (𝐴 +𝑜 𝑥) = suc (𝐴 +𝑜 𝐵))
13	12	uneq2d 3097	. . . . . . . 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 2084	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ suc 𝐵 suc (𝐴 +𝑜 𝑥) = (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ suc (𝐴 +𝑜 𝐵)))
15	14	uneq2d 3097	. . . . . 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 262	. . . . 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 2072	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥) ∪ suc (𝐴 +𝑜 𝐵))))
18		unass 3100	. . . 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 2090	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥)) ∪ suc (𝐴 +𝑜 𝐵)))
20		oav2 6043	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +𝑜 𝑥)))
21	20	uneq1d 3096	. . 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 2075	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = ((𝐴 +𝑜 𝐵) ∪ suc (𝐴 +𝑜 𝐵)))
23		sssucid 4152	. . 3 ⊢ (𝐴 +𝑜 𝐵) ⊆ suc (𝐴 +𝑜 𝐵)
24		ssequn1 3113	. . 3 ⊢ ((𝐴 +𝑜 𝐵) ⊆ suc (𝐴 +𝑜 𝐵) ↔ ((𝐴 +𝑜 𝐵) ∪ suc (𝐴 +𝑜 𝐵)) = suc (𝐴 +𝑜 𝐵))
25	23, 24	mpbi 133	. 2 ⊢ ((𝐴 +𝑜 𝐵) ∪ suc (𝐴 +𝑜 𝐵)) = suc (𝐴 +𝑜 𝐵)
26	22, 25	syl6eq 2088	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ∪ cun 2915   ⊆ wss 2917  {csn 3375  ∪ ciun 3657  Oncon0 4100  suc csuc 4102  (class class class)co 5512   +_𝑜 coa 5998

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005

This theorem is referenced by:  onasuc  6046  nnaordi  6081