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Theorem ltsopi 9670
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 9654 . . . 4 N = (ω ∖ {∅})
2 difss 3721 . . . . 5 (ω ∖ {∅}) ⊆ ω
3 omsson 7031 . . . . 5 ω ⊆ On
42, 3sstri 3597 . . . 4 (ω ∖ {∅}) ⊆ On
51, 4eqsstri 3620 . . 3 N ⊆ On
6 epweon 6945 . . . 4 E We On
7 weso 5075 . . . 4 ( E We On → E Or On)
86, 7ax-mp 5 . . 3 E Or On
9 soss 5023 . . 3 (N ⊆ On → ( E Or On → E Or N))
105, 8, 9mp2 9 . 2 E Or N
11 df-lti 9657 . . . 4 <N = ( E ∩ (N × N))
12 soeq1 5024 . . . 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 5154 . . 3 ( E Or N ↔ ( E ∩ (N × N)) Or N)
1513, 14bitr4i 267 . 2 ( <N Or N ↔ E Or N)
1610, 15mpbir 221 1 <N Or N
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  cdif 3557  cin 3559  wss 3560  c0 3897  {csn 4155   E cep 4993   Or wor 5004   We wwe 5042   × cxp 5082  Oncon0 5692  ωcom 7027  Ncnpi 9626   <N clti 9629
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 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-om 7028  df-ni 9654  df-lti 9657
This theorem is referenced by:  indpi  9689  nqereu  9711  ltsonq  9751  archnq  9762
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