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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 2610 | . 2 ⊢ ∅ = ∅ | |
2 | el1o 7466 | . 2 ⊢ (∅ ∈ 1𝑜 ↔ ∅ = ∅) | |
3 | 1, 2 | mpbir 220 | 1 ⊢ ∅ ∈ 1𝑜 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∅c0 3874 1𝑜c1o 7440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-suc 5646 df-1o 7447 |
This theorem is referenced by: dif20el 7472 oe1m 7512 oen0 7553 oeoa 7564 oeoe 7566 isfin4-3 9020 fin1a2lem4 9108 1lt2pi 9606 indpi 9608 sadcp1 15015 vr1cl2 19384 fvcoe1 19398 vr1cl 19408 subrgvr1cl 19453 coe1mul2lem1 19458 coe1tm 19464 ply1coe 19487 evl1var 19521 evls1var 19523 xkofvcn 21297 pw2f1ocnv 36622 wepwsolem 36630 |
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