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Theorem ltsopi 9748
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 9732 . . . 4 N = (ω ∖ {∅})
2 difss 3770 . . . . 5 (ω ∖ {∅}) ⊆ ω
3 omsson 7111 . . . . 5 ω ⊆ On
42, 3sstri 3645 . . . 4 (ω ∖ {∅}) ⊆ On
51, 4eqsstri 3668 . . 3 N ⊆ On
6 epweon 7025 . . . 4 E We On
7 weso 5134 . . . 4 ( E We On → E Or On)
86, 7ax-mp 5 . . 3 E Or On
9 soss 5082 . . 3 (N ⊆ On → ( E Or On → E Or N))
105, 8, 9mp2 9 . 2 E Or N
11 df-lti 9735 . . . 4 <N = ( E ∩ (N × N))
12 soeq1 5083 . . . 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 5217 . . 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 1523  cdif 3604  cin 3606  wss 3607  c0 3948  {csn 4210   E cep 5057   Or wor 5063   We wwe 5101   × cxp 5141  Oncon0 5761  ωcom 7107  Ncnpi 9704   <N clti 9707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-om 7108  df-ni 9732  df-lti 9735
This theorem is referenced by:  indpi  9767  nqereu  9789  ltsonq  9829  archnq  9840
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