MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oacl Structured version   Visualization version   GIF version

Theorem oacl 7660
Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
Assertion
Ref Expression
oacl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)

Proof of Theorem oacl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . . 4 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
21eleq1d 2715 . . 3 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 ∅) ∈ On))
3 oveq2 6698 . . . 4 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
43eleq1d 2715 . . 3 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 𝑦) ∈ On))
5 oveq2 6698 . . . 4 (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
65eleq1d 2715 . . 3 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 suc 𝑦) ∈ On))
7 oveq2 6698 . . . 4 (𝑥 = 𝐵 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵))
87eleq1d 2715 . . 3 (𝑥 = 𝐵 → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 𝐵) ∈ On))
9 oa0 7641 . . . . 5 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
109eleq1d 2715 . . . 4 (𝐴 ∈ On → ((𝐴 +𝑜 ∅) ∈ On ↔ 𝐴 ∈ On))
1110ibir 257 . . 3 (𝐴 ∈ On → (𝐴 +𝑜 ∅) ∈ On)
12 suceloni 7055 . . . . 5 ((𝐴 +𝑜 𝑦) ∈ On → suc (𝐴 +𝑜 𝑦) ∈ On)
13 oasuc 7649 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
1413eleq1d 2715 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 suc 𝑦) ∈ On ↔ suc (𝐴 +𝑜 𝑦) ∈ On))
1512, 14syl5ibr 236 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 suc 𝑦) ∈ On))
1615expcom 450 . . 3 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 suc 𝑦) ∈ On)))
17 vex 3234 . . . . . 6 𝑥 ∈ V
18 iunon 7481 . . . . . 6 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On) → 𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On)
1917, 18mpan 706 . . . . 5 (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On → 𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On)
20 oalim 7657 . . . . . . 7 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +𝑜 𝑥) = 𝑦𝑥 (𝐴 +𝑜 𝑦))
2117, 20mpanr1 719 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 +𝑜 𝑥) = 𝑦𝑥 (𝐴 +𝑜 𝑦))
2221eleq1d 2715 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 +𝑜 𝑥) ∈ On ↔ 𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On))
2319, 22syl5ibr 236 . . . 4 ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 𝑥) ∈ On))
2423expcom 450 . . 3 (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 𝑥) ∈ On)))
252, 4, 6, 8, 11, 16, 24tfinds3 7106 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +𝑜 𝐵) ∈ On))
2625impcom 445 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  c0 3948   ciun 4552  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690   +𝑜 coa 7602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609
This theorem is referenced by:  omcl  7661  oaord  7672  oacan  7673  oaword  7674  oawordri  7675  oawordeulem  7679  oalimcl  7685  oaass  7686  oaf1o  7688  odi  7704  omopth2  7709  oeoalem  7721  oeoa  7722  oancom  8586  cantnfvalf  8600  dfac12lem2  9004  cdanum  9059  wunex3  9601  rdgeqoa  33348
  Copyright terms: Public domain W3C validator