ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10809

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 10793	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4073	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7817	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3931	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3968	. . 3 ⊢ N ⊆ On
6		epweon 7725	. . . 4 ⊢ E We On
7		weso 5616	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5553	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10796	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5554	. . . 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 5707	. . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N)
15	13, 14	bitr4i 279	. 2 ⊢ ( <_N Or N ↔ E Or N)
16	10, 15	mpbir 232	1 ⊢ <_N Or N

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 = wceq 1547 ∖ cdif 3887 ∩ cin 3889 ⊆ wss 3890 ∅c0 4268 {csn 4562 E cep 5524 Or wor 5532 We wwe 5577 × cxp 5623 Oncon0 6317 ωcom 7813 Ncnpi 10765 <_N clti 10768

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-ext 2712 ax-sep 5225 ax-pr 5369

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-sb 2074 df-clab 2719 df-cleq 2732 df-clel 2815 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-tr 5187 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-ord 6320 df-on 6321 df-om 7814 df-ni 10793 df-lti 10796

This theorem is referenced by: indpi 10828 nqereu 10850 ltsonq 10890 archnq 10901

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