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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 4116 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | prid1 3689 | . 2 ⊢ ∅ ∈ {∅, 1o} |
3 | df2o3 6409 | . 2 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2246 | 1 ⊢ ∅ ∈ 2o |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 ∅c0 3414 {cpr 3584 1oc1o 6388 2oc2o 6389 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-nul 4115 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-nul 3415 df-sn 3589 df-pr 3590 df-suc 4356 df-1o 6395 df-2o 6396 |
This theorem is referenced by: nnnninf 7102 nnnninfeq 7104 fodjuf 7121 mkvprop 7134 nninfwlporlemd 7148 nninfwlporlem 7149 nninfwlpoimlemg 7151 nninfwlpoimlemginf 7152 unct 12397 bj-charfun 13842 bj-charfundc 13843 012of 14028 pwle2 14031 subctctexmid 14034 0nninf 14037 nninfsellemcl 14044 nninffeq 14053 |
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