Theorem onaddscl 28273

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 6847	. . . . 5 ⊢ ( L ‘𝐴) ∈ V
2	1	abrexex 7906	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∈ V
3		fvex 6847	. . . . 5 ⊢ ( L ‘𝐵) ∈ V
4	3	abrexex 7906	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)} ∈ V
5	2, 4	unex 7689	. . 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 27873	. . . . . . . 8 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
8	7	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ No )
9		onno 28251	. . . . . . . 8 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No )
10	9	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝐵 ∈ No )
11	8, 10	addscld 27976	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 +s 𝐵) ∈ No )
12		eleq1 2824	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 𝐵) → (𝑥 ∈ No ↔ (𝑦 +s 𝐵) ∈ No ))
13	11, 12	syl5ibrcom 247	. . . . 5 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑥 = (𝑦 +s 𝐵) → 𝑥 ∈ No ))
14	13	rexlimdva 3137	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵) → 𝑥 ∈ No ))
15	14	abssdv 4019	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ⊆ No )
16		onno 28251	. . . . . . . 8 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
17	16	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → 𝐴 ∈ No )
18		leftno 27873	. . . . . . . 8 ⊢ (𝑦 ∈ ( L ‘𝐵) → 𝑦 ∈ No )
19	18	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → 𝑦 ∈ No )
20	17, 19	addscld 27976	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → (𝐴 +s 𝑦) ∈ No )
21		eleq1 2824	. . . . . 6 ⊢ (𝑥 = (𝐴 +s 𝑦) → (𝑥 ∈ No ↔ (𝐴 +s 𝑦) ∈ No ))
22	20, 21	syl5ibrcom 247	. . . . 5 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) ∧ 𝑦 ∈ ( L ‘𝐵)) → (𝑥 = (𝐴 +s 𝑦) → 𝑥 ∈ No ))
23	22	rexlimdva 3137	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦) → 𝑥 ∈ No ))
24	23	abssdv 4019	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)} ⊆ No )
25	15, 24	unssd 4144	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐴)𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝐵)𝑥 = (𝐴 +s 𝑦)}) ⊆ No )
26		leftssno 27869	. . . . . 6 ⊢ ( L ‘𝐴) ⊆ No
27	1	elpw 4558	. . . . . 6 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
28	26, 27	mpbir 231	. . . . 5 ⊢ ( L ‘𝐴) ∈ 𝒫 No
29		nulsgts 27772	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
30	28, 29	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐴) <<s ∅)
31		leftssno 27869	. . . . . 6 ⊢ ( L ‘𝐵) ⊆ No
32	3	elpw 4558	. . . . . 6 ⊢ (( L ‘𝐵) ∈ 𝒫 No ↔ ( L ‘𝐵) ⊆ No )
33	31, 32	mpbir 231	. . . . 5 ⊢ ( L ‘𝐵) ∈ 𝒫 No
34		nulsgts 27772	. . . . 5 ⊢ (( L ‘𝐵) ∈ 𝒫 No → ( L ‘𝐵) <<s ∅)
35	33, 34	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( L ‘𝐵) <<s ∅)
36		oncutleft 28259	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
37	36	adantr 480	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐴 = (( L ‘𝐴) |s ∅))
38		oncutleft 28259	. . . . 5 ⊢ (𝐵 ∈ Ons → 𝐵 = (( L ‘𝐵) |s ∅))
39	38	adantl 481	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝐵 = (( L ‘𝐵) |s ∅))
40	30, 35, 37, 39	addsunif 27998	. . 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 4312	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)
42	41	abf 4358	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} = ∅
43		rex0 4312	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)
44	43	abf 4358	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)} = ∅
45	42, 44	uneq12i 4118	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = (∅ ∪ ∅)
46		un0 4346	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
47	45, 46	eqtri 2759	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 𝐵)} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝐴 +s 𝑦)}) = ∅
48	47	oveq2i 7369	. . 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 2787	. 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 28255	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) ∈ Ons)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714  ∃wrex 3060  Vcvv 3440   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358   No csur 27607   <<s cslts 27753   |s ccuts 27755   L cleft 27821   +_s cadds 27955  On_scons 28247

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-2o 8398  df-nadd 8594  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec2 27945  df-adds 27956  df-ons 28248

This theorem is referenced by:  addonbday  28275  peano2ons  28276