ltsopi - Metamath Proof Explorer

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Theorem ltsopi 9710

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 9694	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 3737	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7069	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3612	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3635	. . 3 ⊢ N ⊆ On
6		epweon 6983	. . . 4 ⊢ E We On
7		weso 5105	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5053	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 9697	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5054	. . . 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 5183	. . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N)
15	13, 14	bitr4i 267	. 2 ⊢ ( <_N Or N ↔ E Or N)
16	10, 15	mpbir 221	1 ⊢ <_N Or N

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 = wceq 1483 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 E cep 5028 Or wor 5034 We wwe 5072 × cxp 5112 Oncon0 5723 ωcom 7065 Ncnpi 9666 <_N clti 9669

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 df-ni 9694 df-lti 9697

This theorem is referenced by: indpi 9729 nqereu 9751 ltsonq 9791 archnq 9802

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