ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10310

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 10294	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4108	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7584	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3976	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 4001	. . 3 ⊢ N ⊆ On
6		epweon 7497	. . . 4 ⊢ E We On
7		weso 5546	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5493	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10297	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5494	. . . 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 5633	. . 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 1537 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 E cep 5464 Or wor 5473 We wwe 5513 × cxp 5553 Oncon0 6191 ωcom 7580 Ncnpi 10266 <_N clti 10269

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-om 7581 df-ni 10294 df-lti 10297

This theorem is referenced by: indpi 10329 nqereu 10351 ltsonq 10391 archnq 10402

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