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Mirrors > Home > ILE Home > Th. List > df2o3 | GIF version |
Description: Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
df2o3 | ⊢ 2o = {∅, 1o} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 6412 | . 2 ⊢ 2o = suc 1o | |
2 | df-suc 4368 | . 2 ⊢ suc 1o = (1o ∪ {1o}) | |
3 | df1o2 6424 | . . . 4 ⊢ 1o = {∅} | |
4 | 3 | uneq1i 3285 | . . 3 ⊢ (1o ∪ {1o}) = ({∅} ∪ {1o}) |
5 | df-pr 3598 | . . 3 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
6 | 4, 5 | eqtr4i 2201 | . 2 ⊢ (1o ∪ {1o}) = {∅, 1o} |
7 | 1, 2, 6 | 3eqtri 2202 | 1 ⊢ 2o = {∅, 1o} |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∪ cun 3127 ∅c0 3422 {csn 3591 {cpr 3592 suc csuc 4362 1oc1o 6404 2oc2o 6405 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-un 3133 df-nul 3423 df-pr 3598 df-suc 4368 df-1o 6411 df-2o 6412 |
This theorem is referenced by: df2o2 6426 2oconcl 6434 0lt2o 6436 1lt2o 6437 el2oss1o 6438 en2eqpr 6901 nninfisol 7125 finomni 7132 exmidomniim 7133 exmidomni 7134 ismkvnex 7147 nninfwlpoimlemginf 7168 exmidfodomrlemr 7195 exmidfodomrlemrALT 7196 xp2dju 7208 pw1nel3 7224 sucpw1nel3 7226 unct 12423 2o01f 14399 nninfalllem1 14410 nninfall 14411 nninfsellemqall 14417 nninfomnilem 14420 |
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