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Theorem oasuc 6464
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 onsuc 4500 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
2 oav2 6463 . . . . . 6 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)))
31, 2sylan2 286 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)))
4 df-suc 4371 . . . . . . . . . 10 suc 𝐵 = (𝐵 ∪ {𝐵})
5 iuneq1 3899 . . . . . . . . . 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 3966 . . . . . . . . 9 𝑥 ∈ (𝐵 ∪ {𝐵})suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
86, 7eqtri 2198 . . . . . . . 8 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥))
9 oveq2 5882 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
10 suceq 4402 . . . . . . . . . . 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 3962 . . . . . . . . 9 (𝐵 ∈ On → 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
1312uneq2d 3289 . . . . . . . 8 (𝐵 ∈ On → ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥)) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
148, 13eqtrid 2222 . . . . . . 7 (𝐵 ∈ On → 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥) = ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
1514uneq2d 3289 . . . . . 6 (𝐵 ∈ On → (𝐴 𝑥 ∈ suc 𝐵 suc (𝐴 +o 𝑥)) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
1615adantl 277 . . . . 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 2210 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵))))
18 unass 3292 . . . 4 ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)) = (𝐴 ∪ ( 𝑥𝐵 suc (𝐴 +o 𝑥) ∪ suc (𝐴 +o 𝐵)))
1917, 18eqtr4di 2228 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
20 oav2 6463 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)))
2120uneq1d 3288 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = ((𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)) ∪ suc (𝐴 +o 𝐵)))
2219, 21eqtr4d 2213 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)))
23 sssucid 4415 . . 3 (𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵)
24 ssequn1 3305 . . 3 ((𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵) ↔ ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵))
2523, 24mpbi 145 . 2 ((𝐴 +o 𝐵) ∪ suc (𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵)
2622, 25eqtrdi 2226 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  cun 3127  wss 3129  {csn 3592   ciun 3886  Oncon0 4363  suc csuc 4365  (class class class)co 5874   +o coa 6413
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-oadd 6420
This theorem is referenced by:  onasuc  6466  nnaordi  6508
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