Theorem negsex 28194

Description: Every surreal has a negative. Note that this theorem, addscl 28132, addscom 28117, addsass 28156, addsrid 28115, and ltadds1im 28136 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 28187	. 2 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
2		negsid 28192	. 2 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s )
3		oveq2 7408	. . . 4 ⊢ (𝑥 = ( -us ‘𝐴) → (𝐴 +s 𝑥) = (𝐴 +s ( -us ‘𝐴)))
4	3	eqeq1d 2767	. . 3 ⊢ (𝑥 = ( -us ‘𝐴) → ((𝐴 +s 𝑥) = 0s ↔ (𝐴 +s ( -us ‘𝐴)) = 0s ))
5	4	rspcev 3584	. 2 ⊢ ((( -us ‘𝐴) ∈ No ∧ (𝐴 +s ( -us ‘𝐴)) = 0s ) → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s )
6	1, 2, 5	syl2anc 595	1 ⊢ (𝐴 ∈ No → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s )

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  ‘cfv 6525  (class class class)co 7400   No csur 27762   0_s c0s 27956   +_s cadds 28110   -u_s cnegs 28170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec 28089  df-norec2 28100  df-adds 28111  df-negs 28172

This theorem is referenced by: (None)