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Theorem oasuc 6314
 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 4377 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
2 oav2 6313 . . . . . 6 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)))
31, 2sylan2 282 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)))
4 df-suc 4253 . . . . . . . . . 10 suc 𝐵 = (𝐵 ∪ {𝐵})
5 iuneq1 3792 . . . . . . . . . 10 (suc 𝐵 = (𝐵 ∪ {𝐵}) → 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥))
64, 5ax-mp 7 . . . . . . . . 9 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥)
7 iunxun 3858 . . . . . . . . 9 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
86, 7eqtri 2135 . . . . . . . 8 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
9 oveq2 5736 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
10 suceq 4284 . . . . . . . . . . 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 3854 . . . . . . . . 9 (𝐵 ∈ On → 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
1312uneq2d 3196 . . . . . . . 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 2159 . . . . . . 7 (𝐵 ∈ On → 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
1514uneq2d 3196 . . . . . 6 (𝐵 ∈ On → (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
1615adantl 273 . . . . 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 2147 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
18 unass 3199 . . . 4 ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
1917, 18syl6eqr 2165 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
20 oav2 6313 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)))
2120uneq1d 3195 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
2219, 21eqtr4d 2150 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)))
23 sssucid 4297 . . 3 (𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵)
24 ssequn1 3212 . . 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 2163 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 1314   ∈ wcel 1463   ∪ cun 3035   ⊆ wss 3037  {csn 3493  ∪ ciun 3779  Oncon0 4245  suc csuc 4247  (class class class)co 5728   +o coa 6264 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-oadd 6271 This theorem is referenced by:  onasuc  6316  nnaordi  6358
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