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Theorem ltsopi 10928
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 10912 . . . 4 N = (ω ∖ {∅})
2 difss 4136 . . . . 5 (ω ∖ {∅}) ⊆ ω
3 omsson 7891 . . . . 5 ω ⊆ On
42, 3sstri 3993 . . . 4 (ω ∖ {∅}) ⊆ On
51, 4eqsstri 4030 . . 3 N ⊆ On
6 epweon 7795 . . . 4 E We On
7 weso 5676 . . . 4 ( E We On → E Or On)
86, 7ax-mp 5 . . 3 E Or On
9 soss 5612 . . 3 (N ⊆ On → ( E Or On → E Or N))
105, 8, 9mp2 9 . 2 E Or N
11 df-lti 10915 . . . 4 <N = ( E ∩ (N × N))
12 soeq1 5613 . . . 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 5767 . . 3 ( E Or N ↔ ( E ∩ (N × N)) Or N)
1513, 14bitr4i 278 . 2 ( <N Or N ↔ E Or N)
1610, 15mpbir 231 1 <N Or N
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  cdif 3948  cin 3950  wss 3951  c0 4333  {csn 4626   E cep 5583   Or wor 5591   We wwe 5636   × cxp 5683  Oncon0 6384  ωcom 7887  Ncnpi 10884   <N clti 10887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-ord 6387  df-on 6388  df-om 7888  df-ni 10912  df-lti 10915
This theorem is referenced by:  indpi  10947  nqereu  10969  ltsonq  11009  archnq  11020
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