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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 4145 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | prid1 3713 | . 2 ⊢ ∅ ∈ {∅, 1o} |
3 | df2o3 6454 | . 2 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2265 | 1 ⊢ ∅ ∈ 2o |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ∅c0 3437 {cpr 3608 1oc1o 6433 2oc2o 6434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-nul 4144 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-un 3148 df-nul 3438 df-sn 3613 df-pr 3614 df-suc 4389 df-1o 6440 df-2o 6441 |
This theorem is referenced by: nnnninf 7153 nnnninfeq 7155 fodjuf 7172 mkvprop 7185 nninfwlporlemd 7199 nninfwlporlem 7200 nninfwlpoimlemg 7202 nninfwlpoimlemginf 7203 2oneel 7284 2omotaplemst 7286 unct 12492 xpsfeq 12818 xpsfval 12821 xpsval 12825 bj-charfun 15012 bj-charfundc 15013 012of 15199 pwle2 15202 subctctexmid 15204 0nninf 15207 nninfsellemcl 15214 nninffeq 15223 |
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