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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 10294 | . . . 4 ⊢ N = (ω ∖ {∅}) | |
2 | difss 4108 | . . . . 5 ⊢ (ω ∖ {∅}) ⊆ ω | |
3 | omsson 7584 | . . . . 5 ⊢ ω ⊆ On | |
4 | 2, 3 | sstri 3976 | . . . 4 ⊢ (ω ∖ {∅}) ⊆ On |
5 | 1, 4 | eqsstri 4001 | . . 3 ⊢ N ⊆ On |
6 | epweon 7497 | . . . 4 ⊢ E We On | |
7 | weso 5546 | . . . 4 ⊢ ( E We On → E Or On) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ E Or On |
9 | soss 5493 | . . 3 ⊢ (N ⊆ On → ( E Or On → E Or N)) | |
10 | 5, 8, 9 | mp2 9 | . 2 ⊢ E Or N |
11 | df-lti 10297 | . . . 4 ⊢ <N = ( E ∩ (N × N)) | |
12 | soeq1 5494 | . . . 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 5633 | . . 3 ⊢ ( E Or N ↔ ( E ∩ (N × N)) Or N) | |
15 | 13, 14 | bitr4i 280 | . 2 ⊢ ( <N Or N ↔ E Or N) |
16 | 10, 15 | mpbir 233 | 1 ⊢ <N Or N |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 E cep 5464 Or wor 5473 We wwe 5513 × cxp 5553 Oncon0 6191 ωcom 7580 Ncnpi 10266 <N clti 10269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-om 7581 df-ni 10294 df-lti 10297 |
This theorem is referenced by: indpi 10329 nqereu 10351 ltsonq 10391 archnq 10402 |
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