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Theorem oasuc 5983
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 4193 . . . . . 6 (B On → suc B On)
2 oav2 5982 . . . . . 6 ((A On suc B On) → (A +𝑜 suc B) = (A x suc B suc (A +𝑜 x)))
31, 2sylan2 270 . . . . 5 ((A On B On) → (A +𝑜 suc B) = (A x suc B suc (A +𝑜 x)))
4 df-suc 4074 . . . . . . . . . 10 suc B = (B ∪ {B})
5 iuneq1 3661 . . . . . . . . . 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 3726 . . . . . . . . 9 x (B ∪ {B})suc (A +𝑜 x) = ( x B suc (A +𝑜 x) ∪ x {B}suc (A +𝑜 x))
86, 7eqtri 2057 . . . . . . . 8 x suc B suc (A +𝑜 x) = ( x B suc (A +𝑜 x) ∪ x {B}suc (A +𝑜 x))
9 oveq2 5463 . . . . . . . . . . 11 (x = B → (A +𝑜 x) = (A +𝑜 B))
10 suceq 4105 . . . . . . . . . . 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 3723 . . . . . . . . 9 (B On → x {B}suc (A +𝑜 x) = suc (A +𝑜 B))
1312uneq2d 3091 . . . . . . . 8 (B On → ( x B suc (A +𝑜 x) ∪ x {B}suc (A +𝑜 x)) = ( x B suc (A +𝑜 x) ∪ suc (A +𝑜 B)))
148, 13syl5eq 2081 . . . . . . 7 (B On → x suc B suc (A +𝑜 x) = ( x B suc (A +𝑜 x) ∪ suc (A +𝑜 B)))
1514uneq2d 3091 . . . . . 6 (B On → (A x suc B suc (A +𝑜 x)) = (A ∪ ( x B suc (A +𝑜 x) ∪ suc (A +𝑜 B))))
1615adantl 262 . . . . 5 ((A On B On) → (A x suc B suc (A +𝑜 x)) = (A ∪ ( x B suc (A +𝑜 x) ∪ suc (A +𝑜 B))))
173, 16eqtrd 2069 . . . 4 ((A On B On) → (A +𝑜 suc B) = (A ∪ ( x B suc (A +𝑜 x) ∪ suc (A +𝑜 B))))
18 unass 3094 . . . 4 ((A x B suc (A +𝑜 x)) ∪ suc (A +𝑜 B)) = (A ∪ ( x B suc (A +𝑜 x) ∪ suc (A +𝑜 B)))
1917, 18syl6eqr 2087 . . 3 ((A On B On) → (A +𝑜 suc B) = ((A x B suc (A +𝑜 x)) ∪ suc (A +𝑜 B)))
20 oav2 5982 . . . 4 ((A On B On) → (A +𝑜 B) = (A x B suc (A +𝑜 x)))
2120uneq1d 3090 . . 3 ((A On B On) → ((A +𝑜 B) ∪ suc (A +𝑜 B)) = ((A x B suc (A +𝑜 x)) ∪ suc (A +𝑜 B)))
2219, 21eqtr4d 2072 . 2 ((A On B On) → (A +𝑜 suc B) = ((A +𝑜 B) ∪ suc (A +𝑜 B)))
23 sssucid 4118 . . 3 (A +𝑜 B) ⊆ suc (A +𝑜 B)
24 ssequn1 3107 . . 3 ((A +𝑜 B) ⊆ suc (A +𝑜 B) ↔ ((A +𝑜 B) ∪ suc (A +𝑜 B)) = suc (A +𝑜 B))
2523, 24mpbi 133 . 2 ((A +𝑜 B) ∪ suc (A +𝑜 B)) = suc (A +𝑜 B)
2622, 25syl6eq 2085 1 ((A On B On) → (A +𝑜 suc B) = suc (A +𝑜 B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cun 2909  wss 2911  {csn 3367   ciun 3648  Oncon0 4066  suc csuc 4068  (class class class)co 5455   +𝑜 coa 5937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944
This theorem is referenced by:  onasuc  5985  nnaordi  6017
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