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 Description: There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addnidpi (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴)

Proof of Theorem addnidpi
StepHypRef Expression
1 pinn 10292 . . . . 5 (𝐴N𝐴 ∈ ω)
2 elni2 10291 . . . . . 6 (𝐵N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵))
3 nnaordi 8237 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → (𝐴 +o ∅) ∈ (𝐴 +o 𝐵)))
4 nna0 8223 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
54eleq1d 2901 . . . . . . . . . . 11 (𝐴 ∈ ω → ((𝐴 +o ∅) ∈ (𝐴 +o 𝐵) ↔ 𝐴 ∈ (𝐴 +o 𝐵)))
6 nnord 7579 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → Ord 𝐴)
7 ordirr 6206 . . . . . . . . . . . . . 14 (Ord 𝐴 → ¬ 𝐴𝐴)
86, 7syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ ω → ¬ 𝐴𝐴)
9 eleq2 2905 . . . . . . . . . . . . . 14 ((𝐴 +o 𝐵) = 𝐴 → (𝐴 ∈ (𝐴 +o 𝐵) ↔ 𝐴𝐴))
109notbid 319 . . . . . . . . . . . . 13 ((𝐴 +o 𝐵) = 𝐴 → (¬ 𝐴 ∈ (𝐴 +o 𝐵) ↔ ¬ 𝐴𝐴))
118, 10syl5ibrcom 248 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((𝐴 +o 𝐵) = 𝐴 → ¬ 𝐴 ∈ (𝐴 +o 𝐵)))
1211con2d 136 . . . . . . . . . . 11 (𝐴 ∈ ω → (𝐴 ∈ (𝐴 +o 𝐵) → ¬ (𝐴 +o 𝐵) = 𝐴))
135, 12sylbid 241 . . . . . . . . . 10 (𝐴 ∈ ω → ((𝐴 +o ∅) ∈ (𝐴 +o 𝐵) → ¬ (𝐴 +o 𝐵) = 𝐴))
1413adantl 482 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 +o ∅) ∈ (𝐴 +o 𝐵) → ¬ (𝐴 +o 𝐵) = 𝐴))
153, 14syld 47 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → ¬ (𝐴 +o 𝐵) = 𝐴))
1615expcom 414 . . . . . . 7 (𝐴 ∈ ω → (𝐵 ∈ ω → (∅ ∈ 𝐵 → ¬ (𝐴 +o 𝐵) = 𝐴)))
1716imp32 419 . . . . . 6 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) → ¬ (𝐴 +o 𝐵) = 𝐴)
182, 17sylan2b 593 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵N) → ¬ (𝐴 +o 𝐵) = 𝐴)
191, 18sylan 580 . . . 4 ((𝐴N𝐵N) → ¬ (𝐴 +o 𝐵) = 𝐴)
20 addpiord 10298 . . . . 5 ((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
2120eqeq1d 2827 . . . 4 ((𝐴N𝐵N) → ((𝐴 +N 𝐵) = 𝐴 ↔ (𝐴 +o 𝐵) = 𝐴))
2219, 21mtbird 326 . . 3 ((𝐴N𝐵N) → ¬ (𝐴 +N 𝐵) = 𝐴)
2322a1d 25 . 2 ((𝐴N𝐵N) → (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴))
24 dmaddpi 10304 . . . . . 6 dom +N = (N × N)
2524ndmov 7325 . . . . 5 (¬ (𝐴N𝐵N) → (𝐴 +N 𝐵) = ∅)
2625eqeq1d 2827 . . . 4 (¬ (𝐴N𝐵N) → ((𝐴 +N 𝐵) = 𝐴 ↔ ∅ = 𝐴))
27 0npi 10296 . . . . 5 ¬ ∅ ∈ N
28 eleq1 2904 . . . . 5 (∅ = 𝐴 → (∅ ∈ N𝐴N))
2927, 28mtbii 327 . . . 4 (∅ = 𝐴 → ¬ 𝐴N)
3026, 29syl6bi 254 . . 3 (¬ (𝐴N𝐵N) → ((𝐴 +N 𝐵) = 𝐴 → ¬ 𝐴N))
3130con2d 136 . 2 (¬ (𝐴N𝐵N) → (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴))
3223, 31pm2.61i 183 1 (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107  ∅c0 4294  Ord word 6187  (class class class)co 7151  ωcom 7571   +o coa 8093  Ncnpi 10258   +N cpli 10259 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-oadd 8100  df-ni 10286  df-pli 10287 This theorem is referenced by: (None)
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