Theorem oacl 7502

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.

Step	Hyp	Ref	Expression
1		oveq2 6557	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
2	1	eleq1d 2672	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 ∅) ∈ On))
3		oveq2 6557	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
4	3	eleq1d 2672	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 𝑦) ∈ On))
5		oveq2 6557	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
6	5	eleq1d 2672	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 suc 𝑦) ∈ On))
7		oveq2 6557	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵))
8	7	eleq1d 2672	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 +𝑜 𝑥) ∈ On ↔ (𝐴 +𝑜 𝐵) ∈ On))
9		oa0 7483	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
10	9	eleq1d 2672	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 +𝑜 ∅) ∈ On ↔ 𝐴 ∈ On))
11	10	ibir 256	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) ∈ On)
12		suceloni 6905	. . . . 5 ⊢ ((𝐴 +𝑜 𝑦) ∈ On → suc (𝐴 +𝑜 𝑦) ∈ On)
13		oasuc 7491	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
14	13	eleq1d 2672	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 suc 𝑦) ∈ On ↔ suc (𝐴 +𝑜 𝑦) ∈ On))
15	12, 14	syl5ibr 235	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 suc 𝑦) ∈ On))
16	15	expcom 450	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 suc 𝑦) ∈ On)))
17		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
18		iunon 7323	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑦) ∈ On)
19	17, 18	mpan 702	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑦) ∈ On)
20		oalim 7499	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑦))
21	17, 20	mpanr1 715	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑦))
22	21	eleq1d 2672	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 +𝑜 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑦) ∈ On))
23	19, 22	syl5ibr 235	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 𝑥) ∈ On))
24	23	expcom 450	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑦) ∈ On → (𝐴 +𝑜 𝑥) ∈ On)))
25	2, 4, 6, 8, 11, 16, 24	tfinds3 6956	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +𝑜 𝐵) ∈ On))
26	25	impcom 445	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∅c0 3874  ∪ ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +_𝑜 coa 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451

This theorem is referenced by:  omcl  7503  oaord  7514  oacan  7515  oaword  7516  oawordri  7517  oawordeulem  7521  oalimcl  7527  oaass  7528  oaf1o  7530  odi  7546  omopth2  7551  oeoalem  7563  oeoa  7564  oancom  8431  cantnfvalf  8445  dfac12lem2  8849  cdanum  8904  wunex3  9442  rdgeqoa  32394