ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10928

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 10912	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4136	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7891	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3993	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 4030	. . 3 ⊢ N ⊆ On
6		epweon 7795	. . . 4 ⊢ E We On
7		weso 5676	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5612	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10915	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5613	. . . 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 5767	. . 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 1540 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 E cep 5583 Or wor 5591 We wwe 5636 × cxp 5683 Oncon0 6384 ωcom 7887 Ncnpi 10884 <_N clti 10887

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-ord 6387 df-on 6388 df-om 7888 df-ni 10912 df-lti 10915

This theorem is referenced by: indpi 10947 nqereu 10969 ltsonq 11009 archnq 11020

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