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Theorem oasuc 6467
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 4502 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
2 oav2 6466 . . . . . 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 4373 . . . . . . . . . 10 suc 𝐵 = (𝐵 ∪ {𝐵})
5 iuneq1 3901 . . . . . . . . . 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 3968 . . . . . . . . 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 5885 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
10 suceq 4404 . . . . . . . . . . 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 3964 . . . . . . . . 9 (𝐵 ∈ On → 𝑥 ∈ {𝐵}suc (𝐴 +o 𝑥) = suc (𝐴 +o 𝐵))
1312uneq2d 3291 . . . . . . . 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 3291 . . . . . 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 3294 . . . 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 6466 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 𝑥𝐵 suc (𝐴 +o 𝑥)))
2120uneq1d 3290 . . 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 4417 . . 3 (𝐴 +o 𝐵) ⊆ suc (𝐴 +o 𝐵)
24 ssequn1 3307 . . 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 3129  wss 3131  {csn 3594   ciun 3888  Oncon0 4365  suc csuc 4367  (class class class)co 5877   +o coa 6416
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 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423
This theorem is referenced by:  onasuc  6469  nnaordi  6511
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