ltsopi - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ltsopi		Structured version Visualization version GIF version

Theorem ltsopi 10913

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 10897	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4128	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7875	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3986	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 4011	. . 3 ⊢ N ⊆ On
6		epweon 7778	. . . 4 ⊢ E We On
7		weso 5669	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5610	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10900	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5611	. . . 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 5759	. . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N)
15	13, 14	bitr4i 277	. 2 ⊢ ( <_N Or N ↔ E Or N)
16	10, 15	mpbir 230	1 ⊢ <_N Or N

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1533 ∖ cdif 3941 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 {csn 4630 E cep 5581 Or wor 5589 We wwe 5632 × cxp 5676 Oncon0 6371 ωcom 7871 Ncnpi 10869 <_N clti 10872

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-ord 6374 df-on 6375 df-om 7872 df-ni 10897 df-lti 10900

This theorem is referenced by: indpi 10932 nqereu 10954 ltsonq 10994 archnq 11005

W3C validator