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Mirrors > Home > ILE Home > Th. List > 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 4091 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | prid1 3665 | . 2 ⊢ ∅ ∈ {∅, 1o} |
3 | df2o3 6377 | . 2 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2233 | 1 ⊢ ∅ ∈ 2o |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 ∅c0 3394 {cpr 3561 1oc1o 6356 2oc2o 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-nul 4090 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-un 3106 df-nul 3395 df-sn 3566 df-pr 3567 df-suc 4331 df-1o 6363 df-2o 6364 |
This theorem is referenced by: nnnninf 7069 nnnninfeq 7071 fodjuf 7088 mkvprop 7101 unct 12182 bj-charfun 13393 bj-charfundc 13394 012of 13578 pwle2 13581 subctctexmid 13584 0nninf 13587 nninfsellemcl 13594 nninffeq 13603 |
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