ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10883

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 10867	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4132	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7859	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3992	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 4017	. . 3 ⊢ N ⊆ On
6		epweon 7762	. . . 4 ⊢ E We On
7		weso 5668	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5609	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10870	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5610	. . . 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 5758	. . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N)
15	13, 14	bitr4i 278	. 2 ⊢ ( <_N Or N ↔ E Or N)
16	10, 15	mpbir 230	1 ⊢ <_N Or N

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1542 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 {csn 4629 E cep 5580 Or wor 5588 We wwe 5631 × cxp 5675 Oncon0 6365 ωcom 7855 Ncnpi 10839 <_N clti 10842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-ord 6368 df-on 6369 df-om 7856 df-ni 10867 df-lti 10870

This theorem is referenced by: indpi 10902 nqereu 10924 ltsonq 10964 archnq 10975

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