Theorem oacl 8523

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 7422	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2	1	eleq1d 2845	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o ∅) ∈ On))
3		oveq2 7422	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
4	3	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝑦) ∈ On))
5		oveq2 7422	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
6	5	eleq1d 2845	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o suc 𝑦) ∈ On))
7		oveq2 7422	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
8	7	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝐵) ∈ On))
9		oa0 8504	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
10	9	eleq1d 2845	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 +o ∅) ∈ On ↔ 𝐴 ∈ On))
11	10	ibir 271	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) ∈ On)
12		onsuc 7810	. . . . 5 ⊢ ((𝐴 +o 𝑦) ∈ On → suc (𝐴 +o 𝑦) ∈ On)
13		oasuc 8512	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
14	13	eleq1d 2845	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o suc 𝑦) ∈ On ↔ suc (𝐴 +o 𝑦) ∈ On))
15	12, 14	imbitrrid 249	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On))
16	15	expcom 419	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On)))
17		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
18		iunon 8329	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On)
19	17, 18	mpan 703	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On)
20		oalim 8520	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦))
21	17, 20	mpanr1 716	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦))
22	21	eleq1d 2845	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On))
23	19, 22	imbitrrid 249	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On))
24	23	expcom 419	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On)))
25	2, 4, 6, 8, 11, 16, 24	tfinds3 7862	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) ∈ On))
26	25	impcom 413	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  ∅c0 4279  ∪ ciun 4951  Oncon0 6357  Lim wlim 6358  suc csuc 6359  (class class class)co 7414   +_o coa 8453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-oadd 8460

This theorem is used by:  omcl  8524  oaord  8535  oacan  8536  oaword  8537  oawordri  8538  oawordeulem  8542  oalimcl  8548  oaass  8549  oaf1o  8551  odi  8567  omopth2  8572  oeoalem  8585  oeoa  8586  oancom  9631  cantnfvalf  9645  dfac12lem2  10148  djunum  10199  wunex3  10751  rdgeqoa  38125  oaomoecl  44120  oawordex2  44168  omabs2  44174  tfsconcatlem  44178  tfsconcatun  44179  tfsconcatfv2  44182  tfsconcatfv  44183  tfsconcatrn  44184  tfsconcatb0  44186  tfsconcatrev  44190  ofoafg  44196  oaun3lem1  44216  oaun3lem2  44217  oaun3lem3  44218  oaun3lem4  44219  oadif1  44222  oaun2  44223  oaun3  44224  naddgeoa  44236  naddwordnexlem3  44241  oawordex3  44242  naddwordnexlem4  44243  oa1cl  44288