ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10861

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 10845	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4092	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7854	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3948	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3985	. . 3 ⊢ N ⊆ On
6		epweon 7762	. . . 4 ⊢ E We On
7		weso 5643	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5580	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10848	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5581	. . . 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 5734	. . 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

Syntax hints: ↔ wb 209 = wceq 1563 ∖ cdif 3904 ∩ cin 3906 ⊆ wss 3907 ∅c0 4288 {csn 4585 E cep 5551 Or wor 5559 We wwe 5604 × cxp 5650 Oncon0 6350 ωcom 7850 Ncnpi 10817 <_N clti 10820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-ext 2737 ax-sep 5251 ax-pr 5395

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-sb 2094 df-clab 2744 df-cleq 2757 df-clel 2840 df-ne 2961 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-br 5106 df-opab 5168 df-tr 5213 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-ord 6353 df-on 6354 df-om 7851 df-ni 10845 df-lti 10848

This theorem is referenced by: indpi 10880 nqereu 10902 ltsonq 10942 archnq 10953

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