ltsopi - Metamath Proof Explorer

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Theorem ltsopi 11000

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 10984	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4075	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7905	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3922	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3959	. . 3 ⊢ N ⊆ On
6		epweon 7813	. . . 4 ⊢ E We On
7		weso 5646	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5583	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10987	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5584	. . . 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 5738	. . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N)
15	13, 14	bitr4i 281	. 2 ⊢ ( <_N Or N ↔ E Or N)
16	10, 15	mpbir 234	1 ⊢ <_N Or N

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 = wceq 1570 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∅c0 4281 {csn 4588 E cep 5554 Or wor 5562 We wwe 5607 × cxp 5653 Oncon0 6380 ωcom 7901 Ncnpi 10956 <_N clti 10959

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-ext 2735 ax-sep 5252 ax-pr 5395

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-tr 5216 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-ord 6383 df-on 6384 df-om 7902 df-ni 10984 df-lti 10987

This theorem is used by: indpi 11019 nqereu 11041 ltsonq 11081 archnq 11092

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