ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10900

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 10884	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4111	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7863	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3968	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 4005	. . 3 ⊢ N ⊆ On
6		epweon 7767	. . . 4 ⊢ E We On
7		weso 5645	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5581	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10887	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5582	. . . 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 5736	. . 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 231	1 ⊢ <_N Or N

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 = wceq 1540 ∖ cdif 3923 ∩ cin 3925 ⊆ wss 3926 ∅c0 4308 {csn 4601 E cep 5552 Or wor 5560 We wwe 5605 × cxp 5652 Oncon0 6352 ωcom 7859 Ncnpi 10856 <_N clti 10859

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 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2727 df-clel 2809 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-tr 5230 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-ord 6355 df-on 6356 df-om 7860 df-ni 10884 df-lti 10887

This theorem is referenced by: indpi 10919 nqereu 10941 ltsonq 10981 archnq 10992

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