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Theorem ltsopi 9589
Description: Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
Assertion
Ref Expression
ltsopi <N Or N

Proof of Theorem ltsopi
StepHypRef Expression
1 df-ni 9573 . . . 4 N = (ω ∖ {∅})
2 difss 3699 . . . . 5 (ω ∖ {∅}) ⊆ ω
3 omsson 6961 . . . . 5 ω ⊆ On
42, 3sstri 3577 . . . 4 (ω ∖ {∅}) ⊆ On
51, 4eqsstri 3598 . . 3 N ⊆ On
6 epweon 6875 . . . 4 E We On
7 weso 5029 . . . 4 ( E We On → E Or On)
86, 7ax-mp 5 . . 3 E Or On
9 soss 4977 . . 3 (N ⊆ On → ( E Or On → E Or N))
105, 8, 9mp2 9 . 2 E Or N
11 df-lti 9576 . . . 4 <N = ( E ∩ (N × N))
12 soeq1 4978 . . . 4 ( <N = ( E ∩ (N × N)) → ( <N Or N ↔ ( E ∩ (N × N)) Or N))
1311, 12ax-mp 5 . . 3 ( <N Or N ↔ ( E ∩ (N × N)) Or N)
14 soinxp 5106 . . 3 ( E Or N ↔ ( E ∩ (N × N)) Or N)
1513, 14bitr4i 266 . 2 ( <N Or N ↔ E Or N)
1610, 15mpbir 220 1 <N Or N
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  cdif 3537  cin 3539  wss 3540  c0 3874  {csn 4125   E cep 4947   Or wor 4958   We wwe 4996   × cxp 5036  Oncon0 5640  ωcom 6957  Ncnpi 9545   <N clti 9548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958  df-ni 9573  df-lti 9576
This theorem is referenced by:  indpi  9608  nqereu  9630  ltsonq  9670  archnq  9681
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