ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10954

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 10938	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4083	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7870	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3940	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3977	. . 3 ⊢ N ⊆ On
6		epweon 7778	. . . 4 ⊢ E We On
7		weso 5642	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5579	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10941	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5580	. . . 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 5733	. . 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 3896 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 E cep 5550 Or wor 5558 We wwe 5603 × cxp 5649 Oncon0 6355 ωcom 7866 Ncnpi 10910 <_N clti 10913

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 2733 ax-sep 5249 ax-pr 5391

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 2740 df-cleq 2753 df-clel 2836 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-tr 5213 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-ord 6358 df-on 6359 df-om 7867 df-ni 10938 df-lti 10941

This theorem is used by: indpi 10973 nqereu 10995 ltsonq 11035 archnq 11046

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