Theorem nnasmo 8700

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 2761	. . . . 5 ⊢ (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
2		nnacan 8665	. . . . 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 1119	. . 3 ⊢ ((𝐴 ∈ ω ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
5	4	ralrimivva 3200	. 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
6		oveq2 7439	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
7	6	eqeq1d 2737	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑦) = 𝐵))
8	7	rmo4 3739	. 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 1537   ∈ wcel 2106  ∀wral 3059  ∃*wrmo 3377  (class class class)co 7431  ωcom 7887   +_o coa 8502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509

This theorem is referenced by:  ttrcltr  9754