Theorem oacl 8504

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 7404	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2	1	eleq1d 2847	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o ∅) ∈ On))
3		oveq2 7404	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
4	3	eleq1d 2847	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝑦) ∈ On))
5		oveq2 7404	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
6	5	eleq1d 2847	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o suc 𝑦) ∈ On))
7		oveq2 7404	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
8	7	eleq1d 2847	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝐵) ∈ On))
9		oa0 8485	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
10	9	eleq1d 2847	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 +o ∅) ∈ On ↔ 𝐴 ∈ On))
11	10	ibir 270	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) ∈ On)
12		onsuc 7793	. . . . 5 ⊢ ((𝐴 +o 𝑦) ∈ On → suc (𝐴 +o 𝑦) ∈ On)
13		oasuc 8493	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
14	13	eleq1d 2847	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o suc 𝑦) ∈ On ↔ suc (𝐴 +o 𝑦) ∈ On))
15	12, 14	imbitrrid 248	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On))
16	15	expcom 417	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On)))
17		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
18		iunon 8310	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On)
19	17, 18	mpan 700	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On)
20		oalim 8501	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦))
21	17, 20	mpanr1 713	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦))
22	21	eleq1d 2847	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On))
23	19, 22	imbitrrid 248	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On))
24	23	expcom 417	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On)))
25	2, 4, 6, 8, 11, 16, 24	tfinds3 7845	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) ∈ On))
26	25	impcom 411	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  Vcvv 3454  ∅c0 4285  ∪ ciun 4949  Oncon0 6346  Lim wlim 6347  suc csuc 6348  (class class class)co 7396   +_o coa 8434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-oadd 8441

This theorem is referenced by:  omcl  8505  oaord  8516  oacan  8517  oaword  8518  oawordri  8519  oawordeulem  8523  oalimcl  8529  oaass  8530  oaf1o  8532  odi  8548  omopth2  8553  oeoalem  8566  oeoa  8567  oancom  9606  cantnfvalf  9620  dfac12lem2  10101  djunum  10152  wunex3  10699  rdgeqoa  37864  oaomoecl  43855  oawordex2  43903  omabs2  43909  tfsconcatlem  43913  tfsconcatun  43914  tfsconcatfv2  43917  tfsconcatfv  43918  tfsconcatrn  43919  tfsconcatb0  43921  tfsconcatrev  43925  ofoafg  43931  oaun3lem1  43951  oaun3lem2  43952  oaun3lem3  43953  oaun3lem4  43954  oadif1  43957  oaun2  43958  oaun3  43959  naddgeoa  43971  naddwordnexlem3  43976  oawordex3  43977  naddwordnexlem4  43978  oa1cl  44023