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Theorem oasuc 6413
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 4462 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
2 oav2 6412 . . . . . 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 4333 . . . . . . . . . 10 suc 𝐵 = (𝐵 ∪ {𝐵})
5 iuneq1 3864 . . . . . . . . . 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 3930 . . . . . . . . 9 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
86, 7eqtri 2178 . . . . . . . 8 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
9 oveq2 5834 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
10 suceq 4364 . . . . . . . . . . 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 3926 . . . . . . . . 9 (𝐵 ∈ On → 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
1312uneq2d 3262 . . . . . . . 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 2202 . . . . . . 7 (𝐵 ∈ On → 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
1514uneq2d 3262 . . . . . 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 2190 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
18 unass 3265 . . . 4 ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
1917, 18eqtr4di 2208 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
20 oav2 6412 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)))
2120uneq1d 3261 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
2219, 21eqtr4d 2193 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)))
23 sssucid 4377 . . 3 (𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵)
24 ssequn1 3278 . . 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, 25eqtrdi 2206 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 1335  wcel 2128  cun 3100  wss 3102  {csn 3561   ciun 3851  Oncon0 4325  suc csuc 4327  (class class class)co 5826   +o coa 6362
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-oadd 6369
This theorem is referenced by:  onasuc  6415  nnaordi  6457
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