Theorem onaddscl 28312

Description: The surreal ordinals are closed under addition. (Contributed by Scott Fenton, 22-Aug-2025.)

Assertion

Ref	Expression
onaddscl	⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) ∈ Ons)

Proof of Theorem onaddscl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6927	. . . . 5 ⊢ ( L ‘𝐴) ∈ V
2	1	abrexex 7995	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∈ V
3		fvex 6927	. . . . 5 ⊢ ( L ‘𝐵) ∈ V
4	3	abrexex 7995	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)} ∈ V
5	2, 4	unex 7770	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)}) ∈ V
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)}) ∈ V)
7		leftssno 27945	. . . . . . . . 9 ⊢ ( L ‘𝐴) ⊆ No
8	7	sseli 3994	. . . . . . . 8 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
9	8	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ No )
10		onsno 28304	. . . . . . . 8 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No )
11	10	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝐵 ∈ No )
12	9, 11	addscld 28039	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 +s 𝐵) ∈ No )
13		eleq1 2829	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 𝐵) → (𝑥 ∈ No ↔ (𝑦 +s 𝐵) ∈ No ))
14	12, 13	syl5ibrcom 247	. . . . 5 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑥 = (𝑦 +s 𝐵) → 𝑥 ∈ No ))
15	14	rexlimdva 3155	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵) → 𝑥 ∈ No ))
16	15	abssdv 4081	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ⊆ No )
17		onsno 28304	. . . . . . . 8 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
18	17	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → 𝐴 ∈ No )
19		leftssno 27945	. . . . . . . . 9 ⊢ ( L ‘𝐵) ⊆ No
20	19	sseli 3994	. . . . . . . 8 ⊢ (𝑦 ∈ ( L ‘𝐵) → 𝑦 ∈ No )
21	20	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → 𝑦 ∈ No )
22	18, 21	addscld 28039	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → (𝐴 +s 𝑦) ∈ No )
23		eleq1 2829	. . . . . 6 ⊢ (𝑥 = (𝐴 +s 𝑦) → (𝑥 ∈ No ↔ (𝐴 +s 𝑦) ∈ No ))
24	22, 23	syl5ibrcom 247	. . . . 5 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → (𝑥 = (𝐴 +s 𝑦) → 𝑥 ∈ No ))
25	24	rexlimdva 3155	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦) → 𝑥 ∈ No ))
26	25	abssdv 4081	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)} ⊆ No )
27	16, 26	unssd 4205	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)}) ⊆ No )
28	1	elpw 4612	. . . . . 6 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
29	7, 28	mpbir 231	. . . . 5 ⊢ ( L ‘𝐴) ∈ 𝒫 No
30		nulssgt 27869	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
31	29, 30	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐴) <<s ∅)
32	3	elpw 4612	. . . . . 6 ⊢ (( L ‘𝐵) ∈ 𝒫 No ↔ ( L ‘𝐵) ⊆ No )
33	19, 32	mpbir 231	. . . . 5 ⊢ ( L ‘𝐵) ∈ 𝒫 No
34		nulssgt 27869	. . . . 5 ⊢ (( L ‘𝐵) ∈ 𝒫 No → ( L ‘𝐵) <<s ∅)
35	33, 34	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐵) <<s ∅)
36		onscutleft 28311	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
37	36	adantr 480	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = (( L ‘𝐴) |s ∅))
38		onscutleft 28311	. . . . 5 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
39	38	adantl 481	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 = (( L ‘𝐵) |s ∅))
40	31, 35, 37, 39	addsunif 28061	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) = (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)}) |s ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)})))
41		rex0 4369	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)
42	41	abf 4415	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} = ∅
43		rex0 4369	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)
44	43	abf 4415	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)} = ∅
45	42, 44	uneq12i 4179	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = (∅ ∪ ∅)
46		un0 4403	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
47	45, 46	eqtri 2765	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = ∅
48	47	oveq2i 7449	. . 3 ⊢ (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)}) |s ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)})) = (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)}) |s ∅)
49	40, 48	eqtrdi 2793	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) = (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)}) |s ∅))
50	6, 27, 49	elons2d 28308	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) ∈ Ons)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2714  ∃wrex 3070  Vcvv 3481   ∪ cun 3964   ⊆ wss 3966  ∅c0 4342  𝒫 cpw 4608   class class class wbr 5151  ‘cfv 6569  (class class class)co 7438   No csur 27710   <<s csslt 27851   |s cscut 27853   L cleft 27910   +_s cadds 28018  On_scons 28300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-pss 3986  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-tp 4639  df-op 4641  df-ot 4643  df-uni 4916  df-int 4955  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-tr 5269  df-id 5587  df-eprel 5593  df-po 5601  df-so 5602  df-fr 5645  df-se 5646  df-we 5647  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-pred 6329  df-ord 6395  df-on 6396  df-suc 6398  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-riota 7395  df-ov 7441  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023  df-frecs 8314  df-wrecs 8345  df-recs 8419  df-1o 8514  df-2o 8515  df-nadd 8712  df-no 27713  df-slt 27714  df-bday 27715  df-sle 27816  df-sslt 27852  df-scut 27854  df-0s 27895  df-made 27912  df-old 27913  df-left 27915  df-right 27916  df-norec2 28008  df-adds 28019  df-ons 28301

This theorem is referenced by:  peano2ons  28314