|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > df2o2 | Structured version Visualization version GIF version | ||
| Description: Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.) | 
| Ref | Expression | 
|---|---|
| df2o2 | ⊢ 2o = {∅, {∅}} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df2o3 8515 | . 2 ⊢ 2o = {∅, 1o} | |
| 2 | df1o2 8514 | . . 3 ⊢ 1o = {∅} | |
| 3 | 2 | preq2i 4736 | . 2 ⊢ {∅, 1o} = {∅, {∅}} | 
| 4 | 1, 3 | eqtri 2764 | 1 ⊢ 2o = {∅, {∅}} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∅c0 4332 {csn 4625 {cpr 4627 1oc1o 8500 2oc2o 8501 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-un 3955 df-nul 4333 df-sn 4626 df-pr 4628 df-suc 6389 df-1o 8507 df-2o 8508 | 
| This theorem is referenced by: 2dom 9071 pw2eng 9119 pwdju1 10232 canthp1lem1 10693 pr0hash2ex 14448 hashpw 14476 cat1 18143 znidomb 21581 ssoninhaus 36450 onint1 36451 pw2f1ocnv 43054 2omomeqom 43321 df3o3 43332 setc2othin 49138 | 
| Copyright terms: Public domain | W3C validator |