ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10801

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 10785	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4089	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7810	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3947	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3984	. . 3 ⊢ N ⊆ On
6		epweon 7715	. . . 4 ⊢ E We On
7		weso 5614	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5551	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10788	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5552	. . . 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 5705	. . 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 3902 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 {csn 4579 E cep 5522 Or wor 5530 We wwe 5575 × cxp 5621 Oncon0 6311 ωcom 7806 Ncnpi 10757 <_N clti 10760

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 2008 ax-8 2111 ax-9 2119 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-tr 5203 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-ord 6314 df-on 6315 df-om 7807 df-ni 10785 df-lti 10788

This theorem is referenced by: indpi 10820 nqereu 10842 ltsonq 10882 archnq 10893

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