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 10877

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 10861	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4090	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7862	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3946	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3983	. . 3 ⊢ N ⊆ On
6		epweon 7770	. . . 4 ⊢ E We On
7		weso 5652	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5589	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10864	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5590	. . . 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 5743	. . 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 ∖ cdif 3902 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 {csn 4589 E cep 5560 Or wor 5568 We wwe 5613 × cxp 5659 Oncon0 6360 ωcom 7858 Ncnpi 10833 <_N clti 10836

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-ext 2735 ax-sep 5257 ax-pr 5404

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-sb 2097 df-clab 2742 df-cleq 2755 df-clel 2838 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-tr 5219 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-ord 6363 df-on 6364 df-om 7859 df-ni 10861 df-lti 10864

This theorem is used by: indpi 10896 nqereu 10918 ltsonq 10958 archnq 10969

W3C validator