ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10843

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 10827	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4089	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7846	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3945	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3982	. . 3 ⊢ N ⊆ On
6		epweon 7754	. . . 4 ⊢ E We On
7		weso 5636	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5573	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10830	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5574	. . . 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 5727	. . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N)
15	13, 14	bitr4i 280	. 2 ⊢ ( <_N Or N ↔ E Or N)
16	10, 15	mpbir 233	1 ⊢ <_N Or N

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 = wceq 1559 ∖ cdif 3901 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 {csn 4581 E cep 5544 Or wor 5552 We wwe 5597 × cxp 5643 Oncon0 6342 ωcom 7842 Ncnpi 10799 <_N clti 10802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-ext 2733 ax-sep 5245 ax-pr 5389

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-sb 2090 df-clab 2740 df-cleq 2753 df-clel 2836 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-tr 5207 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-ord 6345 df-on 6346 df-om 7843 df-ni 10827 df-lti 10830

This theorem is referenced by: indpi 10862 nqereu 10884 ltsonq 10924 archnq 10935

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