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| 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 8508 | . 2 ⊢ 3o = suc 2o | |
| 2 | df2o2 8515 | . . . 4 ⊢ 2o = {∅, {∅}} | |
| 3 | 2 | sneqi 4637 | . . . 4 ⊢ {2o} = {{∅, {∅}}} |
| 4 | 2, 3 | uneq12i 4166 | . . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}}) |
| 5 | df-suc 6390 | . . 3 ⊢ suc 2o = (2o ∪ {2o}) | |
| 6 | df-tp 4631 | . . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}}) | |
| 7 | 4, 5, 6 | 3eqtr4i 2775 | . 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}} |
| 8 | 1, 7 | eqtri 2765 | 1 ⊢ 3o = {∅, {∅}, {∅, {∅}}} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cun 3949 ∅c0 4333 {csn 4626 {cpr 4628 {ctp 4630 suc csuc 6386 2oc2o 8500 3oc3o 8501 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 df-tp 4631 df-suc 6390 df-1o 8506 df-2o 8507 df-3o 8508 |
| This theorem is referenced by: (None) |
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