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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 6002 | . 2 ⊢ 2𝑜 = suc 1𝑜 | |
2 | df-suc 4108 | . 2 ⊢ suc 1𝑜 = (1𝑜 ∪ {1𝑜}) | |
3 | df1o2 6013 | . . . 4 ⊢ 1𝑜 = {∅} | |
4 | 3 | uneq1i 3093 | . . 3 ⊢ (1𝑜 ∪ {1𝑜}) = ({∅} ∪ {1𝑜}) |
5 | df-pr 3382 | . . 3 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜}) | |
6 | 4, 5 | eqtr4i 2063 | . 2 ⊢ (1𝑜 ∪ {1𝑜}) = {∅, 1𝑜} |
7 | 1, 2, 6 | 3eqtri 2064 | 1 ⊢ 2𝑜 = {∅, 1𝑜} |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∪ cun 2915 ∅c0 3224 {csn 3375 {cpr 3376 suc csuc 4102 1𝑜c1o 5994 2𝑜c2o 5995 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-nul 3225 df-pr 3382 df-suc 4108 df-1o 6001 df-2o 6002 |
This theorem is referenced by: df2o2 6015 2oconcl 6022 |
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