ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10891

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**
Step	Hyp	Ref	Expression
1		df-ni 10875	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4093	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7875	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3949	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3986	. . 3 ⊢ N ⊆ On
6		epweon 7783	. . . 4 ⊢ E We On
7		weso 5657	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5594	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10878	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5595	. . . 4 ⊢ ( <_N = ( E ∩ (N × N)) → ( <_N Or N ↔ ( E ∩ (N × N)) Or N))
13	11, 12	ax-mp 5	. . 3 ⊢ ( <_N Or N ↔ ( E ∩ (N × N)) Or N)
14		soinxp 5748	. . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N)
15	13, 14	bitr4i 281	. 2 ⊢ ( <_N Or N ↔ E Or N)
16	10, 15	mpbir 234	1 ⊢ <_N Or N

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 = wceq 1570 ∖ cdif 3905 ∩ cin 3907 ⊆ wss 3908 ∅c0 4289 {csn 4594 E cep 5565 Or wor 5573 We wwe 5618 × cxp 5664 Oncon0 6367 ωcom 7871 Ncnpi 10847 <_N clti 10850

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-ext 2738 ax-sep 5262 ax-pr 5409

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2745 df-cleq 2758 df-clel 2841 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-tr 5224 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-ord 6370 df-on 6371 df-om 7872 df-ni 10875 df-lti 10878

This theorem is used by: indpi 10910 nqereu 10932 ltsonq 10972 archnq 10983

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