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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   <N clti 10004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-om 7344  df-ni 10029  df-lti 10032
This theorem is referenced by:  indpi  10064  nqereu  10086  ltsonq  10126  archnq  10137
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