ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10776

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 10760	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4086	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7800	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3944	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3981	. . 3 ⊢ N ⊆ On
6		epweon 7708	. . . 4 ⊢ E We On
7		weso 5607	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5544	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10763	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5545	. . . 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 5698	. . 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 1541 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 {csn 4576 E cep 5515 Or wor 5523 We wwe 5568 × cxp 5614 Oncon0 6306 ωcom 7796 Ncnpi 10732 <_N clti 10735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pr 5370

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2710 df-cleq 2723 df-clel 2806 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-tr 5199 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-ord 6309 df-on 6310 df-om 7797 df-ni 10760 df-lti 10763

This theorem is referenced by: indpi 10795 nqereu 10817 ltsonq 10857 archnq 10868

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