Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > df3o3 | Structured version Visualization version GIF version |
Description: Ordinal 3 , fully expanded. (Contributed by RP, 8-Jul-2021.) |
Ref | Expression |
---|---|
df3o3 | ⊢ 3o = {∅, {∅}, {∅, {∅}}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3o 8098 | . 2 ⊢ 3o = suc 2o | |
2 | df2o2 8112 | . . . 4 ⊢ 2o = {∅, {∅}} | |
3 | 2 | sneqi 4572 | . . . 4 ⊢ {2o} = {{∅, {∅}}} |
4 | 2, 3 | uneq12i 4137 | . . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}}) |
5 | df-suc 6192 | . . 3 ⊢ suc 2o = (2o ∪ {2o}) | |
6 | df-tp 4566 | . . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}}) | |
7 | 4, 5, 6 | 3eqtr4i 2854 | . 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}} |
8 | 1, 7 | eqtri 2844 | 1 ⊢ 3o = {∅, {∅}, {∅, {∅}}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3934 ∅c0 4291 {csn 4561 {cpr 4563 {ctp 4565 suc csuc 6188 2oc2o 8090 3oc3o 8091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-dif 3939 df-un 3941 df-nul 4292 df-sn 4562 df-pr 4564 df-tp 4566 df-suc 6192 df-1o 8096 df-2o 8097 df-3o 8098 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |