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Theorem oacl 7560
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 6612 . . . 4 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
21eleq1d 2683 . . 3 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 ∅) ∈ On))
3 oveq2 6612 . . . 4 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
43eleq1d 2683 . . 3 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 𝑦) ∈ On))
5 oveq2 6612 . . . 4 (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
65eleq1d 2683 . . 3 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 suc 𝑦) ∈ On))
7 oveq2 6612 . . . 4 (𝑥 = 𝐵 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵))
87eleq1d 2683 . . 3 (𝑥 = 𝐵 → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 𝐵) ∈ On))
9 oa0 7541 . . . . 5 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
109eleq1d 2683 . . . 4 (𝐴 ∈ On → ((𝐴 +𝑜 ∅) ∈ On ↔ 𝐴 ∈ On))
1110ibir 257 . . 3 (𝐴 ∈ On → (𝐴 +𝑜 ∅) ∈ On)
12 suceloni 6960 . . . . 5 ((𝐴 +𝑜 𝑦) ∈ On → suc (𝐴 +𝑜 𝑦) ∈ On)
13 oasuc 7549 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
1413eleq1d 2683 . . . . 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 451 . . 3 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 suc 𝑦) ∈ On)))
17 vex 3189 . . . . . 6 𝑥 ∈ V
18 iunon 7381 . . . . . 6 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On) → 𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On)
1917, 18mpan 705 . . . . 5 (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On → 𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On)
20 oalim 7557 . . . . . . 7 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +𝑜 𝑥) = 𝑦𝑥 (𝐴 +𝑜 𝑦))
2117, 20mpanr1 718 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 +𝑜 𝑥) = 𝑦𝑥 (𝐴 +𝑜 𝑦))
2221eleq1d 2683 . . . . 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 451 . . 3 (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 𝑥) ∈ On)))
252, 4, 6, 8, 11, 16, 24tfinds3 7011 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +𝑜 𝐵) ∈ On))
2625impcom 446 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  c0 3891   ciun 4485  Oncon0 5682  Lim wlim 5683  suc csuc 5684  (class class class)co 6604   +𝑜 coa 7502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509
This theorem is referenced by:  omcl  7561  oaord  7572  oacan  7573  oaword  7574  oawordri  7575  oawordeulem  7579  oalimcl  7585  oaass  7586  oaf1o  7588  odi  7604  omopth2  7609  oeoalem  7621  oeoa  7622  oancom  8492  cantnfvalf  8506  dfac12lem2  8910  cdanum  8965  wunex3  9507  rdgeqoa  32850
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