ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10628

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 10612	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4070	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7704	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3934	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3959	. . 3 ⊢ N ⊆ On
6		epweon 7616	. . . 4 ⊢ E We On
7		weso 5579	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5522	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10615	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5523	. . . 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 5667	. . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N)
15	13, 14	bitr4i 277	. 2 ⊢ ( <_N Or N ↔ E Or N)
16	10, 15	mpbir 230	1 ⊢ <_N Or N

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1541 ∖ cdif 3888 ∩ cin 3890 ⊆ wss 3891 ∅c0 4261 {csn 4566 E cep 5493 Or wor 5501 We wwe 5542 × cxp 5586 Oncon0 6263 ωcom 7700 Ncnpi 10584 <_N clti 10587

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-11 2157 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7579

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-sb 2071 df-clab 2717 df-cleq 2731 df-clel 2817 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-tr 5196 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-ord 6266 df-on 6267 df-om 7701 df-ni 10612 df-lti 10615

This theorem is referenced by: indpi 10647 nqereu 10669 ltsonq 10709 archnq 10720

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