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Theorem ltsopi 10045
 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 10029 . . . 4 N = (ω ∖ {∅})
2 difss 3960 . . . . 5 (ω ∖ {∅}) ⊆ ω
3 omsson 7347 . . . . 5 ω ⊆ On
42, 3sstri 3830 . . . 4 (ω ∖ {∅}) ⊆ On
51, 4eqsstri 3854 . . 3 N ⊆ On
6 epweon 7260 . . . 4 E We On
7 weso 5346 . . . 4 ( E We On → E Or On)
86, 7ax-mp 5 . . 3 E Or On
9 soss 5293 . . 3 (N ⊆ On → ( E Or On → E Or N))
105, 8, 9mp2 9 . 2 E Or N
11 df-lti 10032 . . . 4 <N = ( E ∩ (N × N))
12 soeq1 5294 . . . 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 5431 . . 3 ( E Or N ↔ ( E ∩ (N × N)) Or N)
1513, 14bitr4i 270 . 2 ( <N Or N ↔ E Or N)
1610, 15mpbir 223 1 <N Or N
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1601   ∖ cdif 3789   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  {csn 4398   E cep 5265   Or wor 5273   We wwe 5313   × cxp 5353  Oncon0 5976  ωcom 7343  Ncnpi 10001
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