ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10901

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 10885	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4086	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7870	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3943	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3980	. . 3 ⊢ N ⊆ On
6		epweon 7778	. . . 4 ⊢ E We On
7		weso 5650	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5587	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10888	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5588	. . . 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 5741	. . 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

This proof depends on syntax axioms: ↔ wb 209 = wceq 1570 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 ∅c0 4282 {csn 4587 E cep 5558 Or wor 5566 We wwe 5611 × cxp 5657 Oncon0 6361 ωcom 7866 Ncnpi 10857 <_N clti 10860

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-ext 2734 ax-sep 5255 ax-pr 5402

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-tr 5217 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-ord 6364 df-on 6365 df-om 7867 df-ni 10885 df-lti 10888

This theorem is used by: indpi 10920 nqereu 10942 ltsonq 10982 archnq 10993

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