ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10299

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 10283	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4059	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7564	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3924	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3949	. . 3 ⊢ N ⊆ On
6		epweon 7477	. . . 4 ⊢ E We On
7		weso 5510	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5457	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10286	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5458	. . . 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 5597	. . 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

Syntax hints: ↔ wb 209 = wceq 1538 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 E cep 5429 Or wor 5437 We wwe 5477 × cxp 5517 Oncon0 6159 ωcom 7560 Ncnpi 10255 <_N clti 10258

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-om 7561 df-ni 10283 df-lti 10286

This theorem is referenced by: indpi 10318 nqereu 10340 ltsonq 10380 archnq 10391

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