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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 2737 | . 2 ⊢ ∅ = ∅ | |
| 2 | el1o 8533 | . 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅) | |
| 3 | 1, 2 | mpbir 231 | 1 ⊢ ∅ ∈ 1o |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∅c0 4333 1oc1o 8499 |
| 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-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-suc 6390 df-1o 8506 |
| This theorem is referenced by: dif20el 8543 oe1m 8583 oen0 8624 oeoa 8635 oeoe 8637 isfin4p1 10355 fin1a2lem4 10443 1lt2pi 10945 indpi 10947 sadcp1 16492 vr1cl2 22194 fvcoe1 22209 vr1cl 22219 subrgvr1cl 22265 coe1mul2lem1 22270 coe1tm 22276 ply1coe 22302 evl1var 22340 evls1var 22342 rhmply1vr1 22391 xkofvcn 23692 pw2f1ocnv 43049 wepwsolem 43054 onexoegt 43256 oaordnrex 43308 omnord1ex 43317 omcl3g 43347 tfsconcatb0 43357 indthinc 49109 indthincALT 49110 prsthinc 49111 |
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