ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10811

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 10795	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4090	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7822	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3945	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3982	. . 3 ⊢ N ⊆ On
6		epweon 7730	. . . 4 ⊢ E We On
7		weso 5623	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5560	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10798	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5561	. . . 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 5714	. . 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 1542 ∖ cdif 3900 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 {csn 4582 E cep 5531 Or wor 5539 We wwe 5584 × cxp 5630 Oncon0 6325 ωcom 7818 Ncnpi 10767 <_N clti 10770

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2709 ax-sep 5243 ax-pr 5379

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-tr 5208 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-ord 6328 df-on 6329 df-om 7819 df-ni 10795 df-lti 10798

This theorem is referenced by: indpi 10830 nqereu 10852 ltsonq 10892 archnq 10903

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