Theorem onaddscl 28451

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 6896	. . . . 5 ⊢ ( L ‘𝐴) ∈ V
2	1	abrexex 7960	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∈ V
3		fvex 6896	. . . . 5 ⊢ ( L ‘𝐵) ∈ V
4	3	abrexex 7960	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)} ∈ V
5	2, 4	unex 7744	. . 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 28051	. . . . . . . 8 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
8	7	adantl 486	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ No )
9		onno 28429	. . . . . . . 8 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No )
10	9	ad2antlr 739	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝐵 ∈ No )
11	8, 10	addscld 28154	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 +s 𝐵) ∈ No )
12		eleq1 2851	. . . . . 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 3166	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵) → 𝑥 ∈ No ))
15	14	abssdv 4022	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ⊆ No )
16		onno 28429	. . . . . . . 8 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
17	16	ad2antrr 738	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → 𝐴 ∈ No )
18		leftno 28051	. . . . . . . 8 ⊢ (𝑦 ∈ ( L ‘𝐵) → 𝑦 ∈ No )
19	18	adantl 486	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → 𝑦 ∈ No )
20	17, 19	addscld 28154	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → (𝐴 +s 𝑦) ∈ No )
21		eleq1 2851	. . . . . 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 3166	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦) → 𝑥 ∈ No ))
24	23	abssdv 4022	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)} ⊆ No )
25	15, 24	unssd 4146	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)}) ⊆ No )
26		leftssno 28047	. . . . . 6 ⊢ ( L ‘𝐴) ⊆ No
27	1	elpw 4567	. . . . . 6 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
28	26, 27	mpbir 234	. . . . 5 ⊢ ( L ‘𝐴) ∈ 𝒫 No
29		nulsgts 27950	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
30	28, 29	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐴) <<s ∅)
31		leftssno 28047	. . . . . 6 ⊢ ( L ‘𝐵) ⊆ No
32	3	elpw 4567	. . . . . 6 ⊢ (( L ‘𝐵) ∈ 𝒫 No ↔ ( L ‘𝐵) ⊆ No )
33	31, 32	mpbir 234	. . . . 5 ⊢ ( L ‘𝐵) ∈ 𝒫 No
34		nulsgts 27950	. . . . 5 ⊢ (( L ‘𝐵) ∈ 𝒫 No → ( L ‘𝐵) <<s ∅)
35	33, 34	mp1i 14	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐵) <<s ∅)
36		oncutleft 28437	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
37	36	adantr 485	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = (( L ‘𝐴) |s ∅))
38		oncutleft 28437	. . . . 5 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
39	38	adantl 486	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 = (( L ‘𝐵) |s ∅))
40	30, 35, 37, 39	addsunif 28176	. . 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 4316	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)
42	41	abf 4372	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} = ∅
43		rex0 4316	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)
44	43	abf 4372	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)} = ∅
45	42, 44	uneq12i 4121	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = (∅ ∪ ∅)
46		un0 4352	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
47	45, 46	eqtri 2786	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = ∅
48	47	oveq2i 7423	. . 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 28433	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) ∈ Ons)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∃wrex 3089  Vcvv 3455   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412   No csur 27785   <<s cslts 27931   |s ccuts 27933   L cleft 27999   +_s cadds 28133  On_scons 28425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27788  df-lts 27789  df-bday 27790  df-les 27890  df-slts 27932  df-cuts 27934  df-0s 27981  df-made 28001  df-old 28002  df-left 28004  df-right 28005  df-norec2 28123  df-adds 28134  df-ons 28426

This theorem is referenced by:  addonbday  28453  peano2ons  28454