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Theorem ltexpi 9668
Description: Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltexpi ((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ ∃𝑥N (𝐴 +N 𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexpi
StepHypRef Expression
1 pinn 9644 . . 3 (𝐴N𝐴 ∈ ω)
2 pinn 9644 . . 3 (𝐵N𝐵 ∈ ω)
3 nnaordex 7663 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
41, 2, 3syl2an 494 . 2 ((𝐴N𝐵N) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
5 ltpiord 9653 . 2 ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))
6 addpiord 9650 . . . . . . 7 ((𝐴N𝑥N) → (𝐴 +N 𝑥) = (𝐴 +𝑜 𝑥))
76eqeq1d 2623 . . . . . 6 ((𝐴N𝑥N) → ((𝐴 +N 𝑥) = 𝐵 ↔ (𝐴 +𝑜 𝑥) = 𝐵))
87pm5.32da 672 . . . . 5 (𝐴N → ((𝑥N ∧ (𝐴 +N 𝑥) = 𝐵) ↔ (𝑥N ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
9 elni2 9643 . . . . . . 7 (𝑥N ↔ (𝑥 ∈ ω ∧ ∅ ∈ 𝑥))
109anbi1i 730 . . . . . 6 ((𝑥N ∧ (𝐴 +𝑜 𝑥) = 𝐵) ↔ ((𝑥 ∈ ω ∧ ∅ ∈ 𝑥) ∧ (𝐴 +𝑜 𝑥) = 𝐵))
11 anass 680 . . . . . 6 (((𝑥 ∈ ω ∧ ∅ ∈ 𝑥) ∧ (𝐴 +𝑜 𝑥) = 𝐵) ↔ (𝑥 ∈ ω ∧ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
1210, 11bitri 264 . . . . 5 ((𝑥N ∧ (𝐴 +𝑜 𝑥) = 𝐵) ↔ (𝑥 ∈ ω ∧ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
138, 12syl6bb 276 . . . 4 (𝐴N → ((𝑥N ∧ (𝐴 +N 𝑥) = 𝐵) ↔ (𝑥 ∈ ω ∧ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
1413rexbidv2 3041 . . 3 (𝐴N → (∃𝑥N (𝐴 +N 𝑥) = 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
1514adantr 481 . 2 ((𝐴N𝐵N) → (∃𝑥N (𝐴 +N 𝑥) = 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
164, 5, 153bitr4d 300 1 ((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ ∃𝑥N (𝐴 +N 𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  c0 3891   class class class wbr 4613  (class class class)co 6604  ωcom 7012   +𝑜 coa 7502  Ncnpi 9610   +N cpli 9611   <N clti 9613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-ni 9638  df-pli 9639  df-lti 9641
This theorem is referenced by:  ltexnq  9741  archnq  9746
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