ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10885

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 10869	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4131	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7861	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3991	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 4016	. . 3 ⊢ N ⊆ On
6		epweon 7764	. . . 4 ⊢ E We On
7		weso 5667	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5608	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10872	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5609	. . . 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 5757	. . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N)
15	13, 14	bitr4i 277	. 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 1541 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {csn 4628 E cep 5579 Or wor 5587 We wwe 5630 × cxp 5674 Oncon0 6364 ωcom 7857 Ncnpi 10841 <_N clti 10844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-ord 6367 df-on 6368 df-om 7858 df-ni 10869 df-lti 10872

This theorem is referenced by: indpi 10904 nqereu 10926 ltsonq 10966 archnq 10977

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