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Theorem addnidpi 9579
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 9556 . . . . 5 (𝐴N𝐴 ∈ ω)
2 elni2 9555 . . . . . 6 (𝐵N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵))
3 nnaordi 7562 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵)))
4 nna0 7548 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
54eleq1d 2671 . . . . . . . . . . 11 (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) ↔ 𝐴 ∈ (𝐴 +𝑜 𝐵)))
6 nnord 6942 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → Ord 𝐴)
7 ordirr 5643 . . . . . . . . . . . . . 14 (Ord 𝐴 → ¬ 𝐴𝐴)
86, 7syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ ω → ¬ 𝐴𝐴)
9 eleq2 2676 . . . . . . . . . . . . . 14 ((𝐴 +𝑜 𝐵) = 𝐴 → (𝐴 ∈ (𝐴 +𝑜 𝐵) ↔ 𝐴𝐴))
109notbid 306 . . . . . . . . . . . . 13 ((𝐴 +𝑜 𝐵) = 𝐴 → (¬ 𝐴 ∈ (𝐴 +𝑜 𝐵) ↔ ¬ 𝐴𝐴))
118, 10syl5ibrcom 235 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((𝐴 +𝑜 𝐵) = 𝐴 → ¬ 𝐴 ∈ (𝐴 +𝑜 𝐵)))
1211con2d 127 . . . . . . . . . . 11 (𝐴 ∈ ω → (𝐴 ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
135, 12sylbid 228 . . . . . . . . . 10 (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
1413adantl 480 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
153, 14syld 45 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
1615expcom 449 . . . . . . 7 (𝐴 ∈ ω → (𝐵 ∈ ω → (∅ ∈ 𝐵 → ¬ (𝐴 +𝑜 𝐵) = 𝐴)))
1716imp32 447 . . . . . 6 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)
182, 17sylan2b 490 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵N) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)
191, 18sylan 486 . . . 4 ((𝐴N𝐵N) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)
20 addpiord 9562 . . . . 5 ((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵))
2120eqeq1d 2611 . . . 4 ((𝐴N𝐵N) → ((𝐴 +N 𝐵) = 𝐴 ↔ (𝐴 +𝑜 𝐵) = 𝐴))
2219, 21mtbird 313 . . 3 ((𝐴N𝐵N) → ¬ (𝐴 +N 𝐵) = 𝐴)
2322a1d 25 . 2 ((𝐴N𝐵N) → (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴))
24 dmaddpi 9568 . . . . . 6 dom +N = (N × N)
2524ndmov 6693 . . . . 5 (¬ (𝐴N𝐵N) → (𝐴 +N 𝐵) = ∅)
2625eqeq1d 2611 . . . 4 (¬ (𝐴N𝐵N) → ((𝐴 +N 𝐵) = 𝐴 ↔ ∅ = 𝐴))
27 0npi 9560 . . . . 5 ¬ ∅ ∈ N
28 eleq1 2675 . . . . 5 (∅ = 𝐴 → (∅ ∈ N𝐴N))
2927, 28mtbii 314 . . . 4 (∅ = 𝐴 → ¬ 𝐴N)
3026, 29syl6bi 241 . . 3 (¬ (𝐴N𝐵N) → ((𝐴 +N 𝐵) = 𝐴 → ¬ 𝐴N))
3130con2d 127 . 2 (¬ (𝐴N𝐵N) → (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴))
3223, 31pm2.61i 174 1 (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  c0 3873  Ord word 5624  (class class class)co 6526  ωcom 6934   +𝑜 coa 7421  Ncnpi 9522   +N cpli 9523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-oadd 7428  df-ni 9550  df-pli 9551
This theorem is referenced by: (None)
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