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Mirrors > Home > ILE Home > Th. List > 1lt2o | GIF version |
Description: Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Ref | Expression |
---|---|
1lt2o | ⊢ 1o ∈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 6329 | . . 3 ⊢ 1o ∈ V | |
2 | 1 | prid2 3638 | . 2 ⊢ 1o ∈ {∅, 1o} |
3 | df2o3 6335 | . 2 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2216 | 1 ⊢ 1o ∈ 2o |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 ∅c0 3368 {cpr 3533 1oc1o 6314 2oc2o 6315 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 df-1o 6321 df-2o 6322 |
This theorem is referenced by: fodjuf 7025 mkvprop 7040 exmidonfinlem 7066 unct 11991 012of 13363 pwle2 13366 subctctexmid 13369 nnsf 13374 peano4nninf 13375 nninfalllemn 13377 nninfsellemcl 13382 nninffeq 13391 |
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