ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10976

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 10960	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4072	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7881	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3917	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3954	. . 3 ⊢ N ⊆ On
6		epweon 7789	. . . 4 ⊢ E We On
7		weso 5641	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5578	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10963	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5579	. . . 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 5732	. . 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 1569 ⊆ wss 3895 ∖ cdif 4045 ∩ cin 4089 ∅c0 4278 {csn 4584 E cep 5549 Or wor 5557 We wwe 5602 × cxp 5648 Oncon0 6362 ωcom 7877 Ncnpi 10932 <_N clti 10935

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-ext 2732 ax-sep 5248 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-sb 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-tr 5212 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-ord 6365 df-on 6366 df-om 7878 df-ni 10960 df-lti 10963

This theorem is used by: indpi 10995 nqereu 11017 ltsonq 11057 archnq 11068

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