Theorem oacl 8450

Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. Remark 2.8 of [Schloeder] p. 5. (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 7354	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2	1	eleq1d 2816	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o ∅) ∈ On))
3		oveq2 7354	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
4	3	eleq1d 2816	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝑦) ∈ On))
5		oveq2 7354	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
6	5	eleq1d 2816	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o suc 𝑦) ∈ On))
7		oveq2 7354	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
8	7	eleq1d 2816	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝐵) ∈ On))
9		oa0 8431	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
10	9	eleq1d 2816	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 +o ∅) ∈ On ↔ 𝐴 ∈ On))
11	10	ibir 268	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) ∈ On)
12		onsuc 7743	. . . . 5 ⊢ ((𝐴 +o 𝑦) ∈ On → suc (𝐴 +o 𝑦) ∈ On)
13		oasuc 8439	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
14	13	eleq1d 2816	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o suc 𝑦) ∈ On ↔ suc (𝐴 +o 𝑦) ∈ On))
15	12, 14	imbitrrid 246	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On))
16	15	expcom 413	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On)))
17		vex 3440	. . . . . 6 ⊢ 𝑥 ∈ V
18		iunon 8259	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On)
19	17, 18	mpan 690	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On)
20		oalim 8447	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦))
21	17, 20	mpanr1 703	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦))
22	21	eleq1d 2816	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On))
23	19, 22	imbitrrid 246	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On))
24	23	expcom 413	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On)))
25	2, 4, 6, 8, 11, 16, 24	tfinds3 7795	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) ∈ On))
26	25	impcom 407	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436  ∅c0 4280  ∪ ciun 4939  Oncon0 6306  Lim wlim 6307  suc csuc 6308  (class class class)co 7346   +_o coa 8382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-oadd 8389

This theorem is referenced by:  omcl  8451  oaord  8462  oacan  8463  oaword  8464  oawordri  8465  oawordeulem  8469  oalimcl  8475  oaass  8476  oaf1o  8478  odi  8494  omopth2  8499  oeoalem  8511  oeoa  8512  oancom  9541  cantnfvalf  9555  dfac12lem2  10036  djunum  10087  wunex3  10632  rdgeqoa  37412  oaomoecl  43319  oawordex2  43367  omabs2  43373  tfsconcatlem  43377  tfsconcatun  43378  tfsconcatfv2  43381  tfsconcatfv  43382  tfsconcatrn  43383  tfsconcatb0  43385  tfsconcatrev  43389  ofoafg  43395  oaun3lem1  43415  oaun3lem2  43416  oaun3lem3  43417  oaun3lem4  43418  oadif1  43421  oaun2  43422  oaun3  43423  naddgeoa  43435  naddwordnexlem3  43440  oawordex3  43441  naddwordnexlem4  43442  oa1cl  43488