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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 6307 | . 2 ⊢ 2o = suc 1o | |
2 | df-suc 4288 | . 2 ⊢ suc 1o = (1o ∪ {1o}) | |
3 | df1o2 6319 | . . . 4 ⊢ 1o = {∅} | |
4 | 3 | uneq1i 3221 | . . 3 ⊢ (1o ∪ {1o}) = ({∅} ∪ {1o}) |
5 | df-pr 3529 | . . 3 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
6 | 4, 5 | eqtr4i 2161 | . 2 ⊢ (1o ∪ {1o}) = {∅, 1o} |
7 | 1, 2, 6 | 3eqtri 2162 | 1 ⊢ 2o = {∅, 1o} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∪ cun 3064 ∅c0 3358 {csn 3522 {cpr 3523 suc csuc 4282 1oc1o 6299 2oc2o 6300 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-un 3070 df-nul 3359 df-pr 3529 df-suc 4288 df-1o 6306 df-2o 6307 |
This theorem is referenced by: df2o2 6321 2oconcl 6329 0lt2o 6331 1lt2o 6332 en2eqpr 6794 finomni 7005 exmidomniim 7006 exmidomni 7007 ismkvnex 7022 exmidfodomrlemr 7051 exmidfodomrlemrALT 7052 xp2dju 7064 unct 11943 el2oss1o 13177 nninfalllem1 13192 nninfall 13193 nninfsellemqall 13200 nninfomnilem 13203 isomninnlem 13214 |
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