ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10786

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 10770	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4085	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7806	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3940	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3977	. . 3 ⊢ N ⊆ On
6		epweon 7714	. . . 4 ⊢ E We On
7		weso 5610	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5547	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10773	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5548	. . . 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 5701	. . 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 3895 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 {csn 4575 E cep 5518 Or wor 5526 We wwe 5571 × cxp 5617 Oncon0 6311 ωcom 7802 Ncnpi 10742 <_N clti 10745

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 2115 ax-9 2123 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pr 5372

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 2712 df-cleq 2725 df-clel 2808 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-tr 5201 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-ord 6314 df-on 6315 df-om 7803 df-ni 10770 df-lti 10773

This theorem is referenced by: indpi 10805 nqereu 10827 ltsonq 10867 archnq 10878

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