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Mirrors > Home > MPE Home > Th. List > 0lt1o | Structured version Visualization version GIF version |
Description: Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.) |
Ref | Expression |
---|---|
0lt1o | ⊢ ∅ ∈ 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . 2 ⊢ ∅ = ∅ | |
2 | el1o 8525 | . 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅) | |
3 | 1, 2 | mpbir 230 | 1 ⊢ ∅ ∈ 1o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ∅c0 4325 1oc1o 8489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5311 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-dif 3950 df-un 3952 df-nul 4326 df-sn 4634 df-suc 6382 df-1o 8496 |
This theorem is referenced by: dif20el 8535 oe1m 8575 oen0 8616 oeoa 8627 oeoe 8629 isfin4p1 10358 fin1a2lem4 10446 1lt2pi 10948 indpi 10950 sadcp1 16455 vr1cl2 22182 fvcoe1 22197 vr1cl 22207 subrgvr1cl 22253 coe1mul2lem1 22258 coe1tm 22264 ply1coe 22289 evl1var 22327 evls1var 22329 rhmply1vr1 22378 xkofvcn 23679 pw2f1ocnv 42695 wepwsolem 42703 onexoegt 42909 oaordnrex 42961 omnord1ex 42970 omcl3g 43000 tfsconcatb0 43010 indthinc 48373 indthincALT 48374 prsthinc 48375 |
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