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Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > 0lt2o | GIF version |
Description: Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Ref | Expression |
---|---|
0lt2o | ⊢ ∅ ∈ 2𝑜 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3966 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | prid1 3548 | . 2 ⊢ ∅ ∈ {∅, 1𝑜} |
3 | df2o3 6195 | . 2 ⊢ 2𝑜 = {∅, 1𝑜} | |
4 | 2, 3 | eleqtrri 2163 | 1 ⊢ ∅ ∈ 2𝑜 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 ∅c0 3286 {cpr 3447 1𝑜c1o 6174 2𝑜c2o 6175 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-nul 3965 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-dif 3001 df-un 3003 df-nul 3287 df-sn 3452 df-pr 3453 df-suc 4198 df-1o 6181 df-2o 6182 |
This theorem is referenced by: 0nninf 11893 nninfalllemn 11898 nninfsellemcl 11903 |
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