Theorem negsex 28076

Description: Every surreal has a negative. Note that this theorem, addscl 28015, addscom 28000, addsass 28039, addsrid 27998, and sltadd1im 28019 are the ordered Abelian group axioms. However, the surreals cannot be said to be an ordered Abelian group because No is a proper class. (Contributed by Scott Fenton, 3-Feb-2025.)

Assertion

Ref	Expression
negsex	⊢ (𝐴 ∈ No → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s )

Distinct variable group:   𝑥,𝐴

Proof of Theorem negsex

Step	Hyp	Ref	Expression
1		negscl 28069	. 2 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
2		negsid 28074	. 2 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s )
3		oveq2 7440	. . . 4 ⊢ (𝑥 = ( -us ‘𝐴) → (𝐴 +s 𝑥) = (𝐴 +s ( -us ‘𝐴)))
4	3	eqeq1d 2738	. . 3 ⊢ (𝑥 = ( -us ‘𝐴) → ((𝐴 +s 𝑥) = 0s ↔ (𝐴 +s ( -us ‘𝐴)) = 0s ))
5	4	rspcev 3621	. 2 ⊢ ((( -us ‘𝐴) ∈ No ∧ (𝐴 +s ( -us ‘𝐴)) = 0s ) → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s )
6	1, 2, 5	syl2anc 584	1 ⊢ (𝐴 ∈ No → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s )

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ∃wrex 3069  ‘cfv 6560  (class class class)co 7432   No csur 27685   0_s c0s 27868   +_s cadds 27993   -u_s cnegs 28052

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 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054

This theorem is referenced by: (None)