Theorem nnasmo 8680

Description: There is at most one left additive inverse for natural number addition. (Contributed by Scott Fenton, 17-Oct-2024.)

Assertion

Ref	Expression
nnasmo	⊢ (𝐴 ∈ ω → ∃*𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnasmo

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2758	. . . . 5 ⊢ (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
2		nnacan 8645	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
3	1, 2	imbitrid 244	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
4	3	3expb 1120	. . 3 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
5	4	ralrimivva 3188	. 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
6		oveq2 7418	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
7	6	eqeq1d 2738	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑦) = 𝐵))
8	7	rmo4 3718	. 2 ⊢ (∃*𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵 ↔ ∀𝑥 ∈ ω ∀𝑦 ∈ ω (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
9	5, 8	sylibr 234	1 ⊢ (𝐴 ∈ ω → ∃*𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃*wrmo 3363  (class class class)co 7410  ωcom 7866   +_o coa 8482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-oadd 8489

This theorem is referenced by:  ttrcltr  9735