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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 10912 | . . . 4 ⊢ N = (ω ∖ {∅}) | |
| 2 | difss 4136 | . . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω | |
| 3 | omsson 7891 | . . . . 5 ⊢ ω ⊆ On | |
| 4 | 2, 3 | sstri 3993 | . . . 4 ⊢ (ω ∖ {∅}) ⊆ On |
| 5 | 1, 4 | eqsstri 4030 | . . 3 ⊢ N ⊆ On |
| 6 | epweon 7795 | . . . 4 ⊢ E We On | |
| 7 | weso 5676 | . . . 4 ⊢ ( E We On → E Or On) | |
| 8 | 6, 7 | ax-mp 5 | . . 3 ⊢ E Or On |
| 9 | soss 5612 | . . 3 ⊢ (N ⊆ On → ( E Or On → E Or N)) | |
| 10 | 5, 8, 9 | mp2 9 | . 2 ⊢ E Or N |
| 11 | df-lti 10915 | . . . 4 ⊢ <N = ( E ∩ (N × N)) | |
| 12 | soeq1 5613 | . . . 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 5767 | . . 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 231 | 1 ⊢ <N Or N |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 E cep 5583 Or wor 5591 We wwe 5636 × cxp 5683 Oncon0 6384 ωcom 7887 Ncnpi 10884 <N clti 10887 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-ord 6387 df-on 6388 df-om 7888 df-ni 10912 df-lti 10915 |
| This theorem is referenced by: indpi 10947 nqereu 10969 ltsonq 11009 archnq 11020 |
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