Theorem onaddscl 28208

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 6835	. . . . 5 ⊢ ( L ‘𝐴) ∈ V
2	1	abrexex 7894	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∈ V
3		fvex 6835	. . . . 5 ⊢ ( L ‘𝐵) ∈ V
4	3	abrexex 7894	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)} ∈ V
5	2, 4	unex 7677	. . 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 27824	. . . . . . . . 9 ⊢ ( L ‘𝐴) ⊆ No
8	7	sseli 3930	. . . . . . . 8 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
9	8	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ No )
10		onsno 28190	. . . . . . . 8 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No )
11	10	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝐵 ∈ No )
12	9, 11	addscld 27921	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 +s 𝐵) ∈ No )
13		eleq1 2819	. . . . . 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 3133	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵) → 𝑥 ∈ No ))
16	15	abssdv 4019	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ⊆ No )
17		onsno 28190	. . . . . . . 8 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
18	17	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → 𝐴 ∈ No )
19		leftssno 27824	. . . . . . . . 9 ⊢ ( L ‘𝐵) ⊆ No
20	19	sseli 3930	. . . . . . . 8 ⊢ (𝑦 ∈ ( L ‘𝐵) → 𝑦 ∈ No )
21	20	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → 𝑦 ∈ No )
22	18, 21	addscld 27921	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → (𝐴 +s 𝑦) ∈ No )
23		eleq1 2819	. . . . . 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 3133	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦) → 𝑥 ∈ No ))
26	25	abssdv 4019	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)} ⊆ No )
27	16, 26	unssd 4142	. 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 4554	. . . . . 6 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
29	7, 28	mpbir 231	. . . . 5 ⊢ ( L ‘𝐴) ∈ 𝒫 No
30		nulssgt 27737	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
31	29, 30	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐴) <<s ∅)
32	3	elpw 4554	. . . . . 6 ⊢ (( L ‘𝐵) ∈ 𝒫 No ↔ ( L ‘𝐵) ⊆ No )
33	19, 32	mpbir 231	. . . . 5 ⊢ ( L ‘𝐵) ∈ 𝒫 No
34		nulssgt 27737	. . . . 5 ⊢ (( L ‘𝐵) ∈ 𝒫 No → ( L ‘𝐵) <<s ∅)
35	33, 34	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐵) <<s ∅)
36		onscutleft 28198	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
37	36	adantr 480	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = (( L ‘𝐴) |s ∅))
38		onscutleft 28198	. . . . 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 27943	. . 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 4310	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)
42	41	abf 4356	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} = ∅
43		rex0 4310	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)
44	43	abf 4356	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)} = ∅
45	42, 44	uneq12i 4116	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = (∅ ∪ ∅)
46		un0 4344	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
47	45, 46	eqtri 2754	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = ∅
48	47	oveq2i 7357	. . 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 2782	. 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 28194	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) ∈ Ons)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056  Vcvv 3436   ∪ cun 3900   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346   No csur 27576   <<s csslt 27718   |s cscut 27720   L cleft 27784   +_s cadds 27900  On_scons 28186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-ot 4585  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-nadd 8581  df-no 27579  df-slt 27580  df-bday 27581  df-sle 27682  df-sslt 27719  df-scut 27721  df-0s 27766  df-made 27786  df-old 27787  df-left 27789  df-right 27790  df-norec2 27890  df-adds 27901  df-ons 28187

This theorem is referenced by:  peano2ons  28210