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Mirrors > Home > MPE Home > Th. List > ltsopi | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ltsopi | ⊢ <N Or N |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ni 10678 | . . . 4 ⊢ N = (ω ∖ {∅}) | |
2 | difss 4072 | . . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω | |
3 | omsson 7748 | . . . . 5 ⊢ ω ⊆ On | |
4 | 2, 3 | sstri 3935 | . . . 4 ⊢ (ω ∖ {∅}) ⊆ On |
5 | 1, 4 | eqsstri 3960 | . . 3 ⊢ N ⊆ On |
6 | epweon 7657 | . . . 4 ⊢ E We On | |
7 | weso 5591 | . . . 4 ⊢ ( E We On → E Or On) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ E Or On |
9 | soss 5534 | . . 3 ⊢ (N ⊆ On → ( E Or On → E Or N)) | |
10 | 5, 8, 9 | mp2 9 | . 2 ⊢ E Or N |
11 | df-lti 10681 | . . . 4 ⊢ <N = ( E ∩ (N × N)) | |
12 | soeq1 5535 | . . . 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 5679 | . . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N) | |
15 | 13, 14 | bitr4i 278 | . 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 1539 ∖ cdif 3889 ∩ cin 3891 ⊆ wss 3892 ∅c0 4262 {csn 4565 E cep 5505 Or wor 5513 We wwe 5554 × cxp 5598 Oncon0 6281 ωcom 7744 Ncnpi 10650 <N clti 10653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3306 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-tr 5199 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-ord 6284 df-on 6285 df-om 7745 df-ni 10678 df-lti 10681 |
This theorem is referenced by: indpi 10713 nqereu 10735 ltsonq 10775 archnq 10786 |
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