Theorem onaddscl 28709

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 8006	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ (L‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∈ V
3		fvex 6927	. . . . 5 ⊢ (L‘𝐵) ∈ V
4	3	abrexex 8006	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ (L‘𝐵)𝑥 = (𝐴 +s 𝑦)} ∈ V
5	2, 4	unex 7791	. . 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		leftno 28309	. . . . . . . 8 ⊢ (𝑦 ∈ (L‘𝐴) → 𝑦 ∈ No)
8	7	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ (L‘𝐴)) → 𝑦 ∈ No)
9		onno 28687	. . . . . . . 8 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No)
10	9	ad2antlr 740	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ (L‘𝐴)) → 𝐵 ∈ No)
11	8, 10	addscld 28412	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ (L‘𝐴)) → (𝑦 +s 𝐵) ∈ No)
12		eleq1 2852	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 𝐵) → (𝑥 ∈ No ↔ (𝑦 +s 𝐵) ∈ No))
13	11, 12	syl5ibrcom 250	. . . . 5 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ (L‘𝐴)) → (𝑥 = (𝑦 +s 𝐵) → 𝑥 ∈ No))
14	13	rexlimdva 3167	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ (L‘𝐴)𝑥 = (𝑦 +s 𝐵) → 𝑥 ∈ No))
15	14	abssdv 3995	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ (L‘𝐴)𝑥 = (𝑦 +s 𝐵)} ⊆ No)
16		onno 28687	. . . . . . . 8 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No)
17	16	ad2antrr 739	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ (L‘𝐵)) → 𝐴 ∈ No)
18		leftno 28309	. . . . . . . 8 ⊢ (𝑦 ∈ (L‘𝐵) → 𝑦 ∈ No)
19	18	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ (L‘𝐵)) → 𝑦 ∈ No)
20	17, 19	addscld 28412	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ (L‘𝐵)) → (𝐴 +s 𝑦) ∈ No)
21		eleq1 2852	. . . . . 6 ⊢ (𝑥 = (𝐴 +s 𝑦) → (𝑥 ∈ No ↔ (𝐴 +s 𝑦) ∈ No))
22	20, 21	syl5ibrcom 250	. . . . 5 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ (L‘𝐵)) → (𝑥 = (𝐴 +s 𝑦) → 𝑥 ∈ No))
23	22	rexlimdva 3167	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ (L‘𝐵)𝑥 = (𝐴 +s 𝑦) → 𝑥 ∈ No))
24	23	abssdv 3995	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ (L‘𝐵)𝑥 = (𝐴 +s 𝑦)} ⊆ No)
25	15, 24	unssd 4193	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ({𝑥 ∣ ∃𝑦 ∈ (L‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ (L‘𝐵)𝑥 = (𝐴 +s 𝑦)}) ⊆ No)
26		leftssno 28305	. . . . . 6 ⊢ (L‘𝐴) ⊆ No
27	1	elpw 4564	. . . . . 6 ⊢ ((L‘𝐴) ∈ 𝒫 No ↔ (L‘𝐴) ⊆ No)
28	26, 27	mpbir 234	. . . . 5 ⊢ (L‘𝐴) ∈ 𝒫 No
29		nulsgts 28208	. . . . 5 ⊢ ((L‘𝐴) ∈ 𝒫 No → (L‘𝐴) <<s ∅)
30	28, 29	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (L‘𝐴) <<s ∅)
31		leftssno 28305	. . . . . 6 ⊢ (L‘𝐵) ⊆ No
32	3	elpw 4564	. . . . . 6 ⊢ ((L‘𝐵) ∈ 𝒫 No ↔ (L‘𝐵) ⊆ No)
33	31, 32	mpbir 234	. . . . 5 ⊢ (L‘𝐵) ∈ 𝒫 No
34		nulsgts 28208	. . . . 5 ⊢ ((L‘𝐵) ∈ 𝒫 No → (L‘𝐵) <<s ∅)
35	33, 34	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (L‘𝐵) <<s ∅)
36		oncutleft 28695	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = ((L‘𝐴) |s ∅))
37	36	adantr 486	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = ((L‘𝐴) |s ∅))
38		oncutleft 28695	. . . . 5 ⊢ (𝐵 ∈ Ons → 𝐵 = ((L‘𝐵) |s ∅))
39	38	adantl 487	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 = ((L‘𝐵) |s ∅))
40	30, 35, 37, 39	addsunif 28434	. . 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 4311	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)
42	41	abf 4367	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} = ∅
43		rex0 4311	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)
44	43	abf 4367	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)} = ∅
45	42, 44	uneq12i 4168	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = (∅ ∪ ∅)
46		un0 4347	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
47	45, 46	eqtri 2786	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = ∅
48	47	oveq2i 7460	. . 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 2814	. 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, 25, 49	elons2d 28691	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) ∈ Ons)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560   class class class wbr 5106  ‘cfv 6564  (class class class)co 7449  Nocsur 28043   <<s cslts 28189   |s ccuts 28191  Lcleft 28257   +_s cadds 28391  On_scons 28683

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-1o 8512  df-2o 8513  df-nadd 8711  df-no 28046  df-lts 28047  df-bday 28048  df-les 28148  df-slts 28190  df-cuts 28192  df-0s 28239  df-made 28259  df-old 28260  df-left 28262  df-right 28263  df-norec2 28381  df-adds 28392  df-ons 28684

This theorem is used by:  addonbday  28711  peano2ons  28712