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Theorem oasuc 6328
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.
StepHypRef Expression
1 suceloni 4387 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
2 oav2 6327 . . . . . 6 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)))
31, 2sylan2 284 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)))
4 df-suc 4263 . . . . . . . . . 10 suc 𝐵 = (𝐵 ∪ {𝐵})
5 iuneq1 3796 . . . . . . . . . 10 (suc 𝐵 = (𝐵 ∪ {𝐵}) → 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥))
64, 5ax-mp 5 . . . . . . . . 9 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥)
7 iunxun 3862 . . . . . . . . 9 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
86, 7eqtri 2138 . . . . . . . 8 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
9 oveq2 5750 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
10 suceq 4294 . . . . . . . . . . 11 ((𝐴 +o 𝑥) = (𝐴 +o 𝐵) → suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
119, 10syl 14 . . . . . . . . . 10 (𝑥 = 𝐵 → suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
1211iunxsng 3858 . . . . . . . . 9 (𝐵 ∈ On → 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
1312uneq2d 3200 . . . . . . . 8 (𝐵 ∈ On → ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥)) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
148, 13syl5eq 2162 . . . . . . 7 (𝐵 ∈ On → 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
1514uneq2d 3200 . . . . . 6 (𝐵 ∈ On → (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
1615adantl 275 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
173, 16eqtrd 2150 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
18 unass 3203 . . . 4 ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
1917, 18syl6eqr 2168 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
20 oav2 6327 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)))
2120uneq1d 3199 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
2219, 21eqtr4d 2153 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)))
23 sssucid 4307 . . 3 (𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵)
24 ssequn1 3216 . . 3 ((𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵) ↔ ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵))
2523, 24mpbi 144 . 2 ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵)
2622, 25syl6eq 2166 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  cun 3039  wss 3041  {csn 3497   ciun 3783  Oncon0 4255  suc csuc 4257  (class class class)co 5742   +o coa 6278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285
This theorem is referenced by:  onasuc  6330  nnaordi  6372
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