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Theorem ltsopi 10308
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 10292 . . . 4 N = (ω ∖ {∅})
2 difss 4094 . . . . 5 (ω ∖ {∅}) ⊆ ω
3 omsson 7578 . . . . 5 ω ⊆ On
42, 3sstri 3962 . . . 4 (ω ∖ {∅}) ⊆ On
51, 4eqsstri 3987 . . 3 N ⊆ On
6 epweon 7491 . . . 4 E We On
7 weso 5533 . . . 4 ( E We On → E Or On)
86, 7ax-mp 5 . . 3 E Or On
9 soss 5480 . . 3 (N ⊆ On → ( E Or On → E Or N))
105, 8, 9mp2 9 . 2 E Or N
11 df-lti 10295 . . . 4 <N = ( E ∩ (N × N))
12 soeq1 5481 . . . 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 5620 . . 3 ( E Or N ↔ ( E ∩ (N × N)) Or N)
1513, 14bitr4i 281 . 2 ( <N Or N ↔ E Or N)
1610, 15mpbir 234 1 <N Or N
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  cdif 3916  cin 3918  wss 3919  c0 4276  {csn 4550   E cep 5451   Or wor 5460   We wwe 5500   × cxp 5540  Oncon0 6178  ωcom 7574  Ncnpi 10264   <N clti 10267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-om 7575  df-ni 10292  df-lti 10295
This theorem is referenced by:  indpi  10327  nqereu  10349  ltsonq  10389  archnq  10400
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