ltsopi - Metamath Proof Explorer

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Theorem ltsopi 10874

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 10858	. . . 4 ⊢ N = (ω ∖ {∅})
2		difss 4091	. . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω
3		omsson 7867	. . . . 5 ⊢ ω ⊆ On
4	2, 3	sstri 3947	. . . 4 ⊢ (ω ∖ {∅}) ⊆ On
5	1, 4	eqsstri 3984	. . 3 ⊢ N ⊆ On
6		epweon 7775	. . . 4 ⊢ E We On
7		weso 5654	. . . 4 ⊢ ( E We On → E Or On)
8	6, 7	ax-mp 5	. . 3 ⊢ E Or On
9		soss 5591	. . 3 ⊢ (N ⊆ On → ( E Or On → E Or N))
10	5, 8, 9	mp2 9	. 2 ⊢ E Or N
11		df-lti 10861	. . . 4 ⊢ <_N = ( E ∩ (N × N))
12		soeq1 5592	. . . 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 5745	. . 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 1570 ∖ cdif 3903 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 {csn 4590 E cep 5562 Or wor 5570 We wwe 5615 × cxp 5661 Oncon0 6362 ωcom 7863 Ncnpi 10830 <_N clti 10833

This theorem was proved from 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 5258 ax-pr 5406

This theorem 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 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-tr 5220 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-ord 6365 df-on 6366 df-om 7864 df-ni 10858 df-lti 10861

This theorem is referenced by: indpi 10893 nqereu 10915 ltsonq 10955 archnq 10966

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