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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 7449 | . 2 ⊢ 3𝑜 = suc 2𝑜 | |
2 | df2o2 7461 | . . . 4 ⊢ 2𝑜 = {∅, {∅}} | |
3 | 2 | sneqi 4136 | . . . 4 ⊢ {2𝑜} = {{∅, {∅}}} |
4 | 2, 3 | uneq12i 3727 | . . 3 ⊢ (2𝑜 ∪ {2𝑜}) = ({∅, {∅}} ∪ {{∅, {∅}}}) |
5 | df-suc 5646 | . . 3 ⊢ suc 2𝑜 = (2𝑜 ∪ {2𝑜}) | |
6 | df-tp 4130 | . . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}}) | |
7 | 4, 5, 6 | 3eqtr4i 2642 | . 2 ⊢ suc 2𝑜 = {∅, {∅}, {∅, {∅}}} |
8 | 1, 7 | eqtri 2632 | 1 ⊢ 3𝑜 = {∅, {∅}, {∅, {∅}}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 ∅c0 3874 {csn 4125 {cpr 4127 {ctp 4129 suc csuc 5642 2𝑜c2o 7441 3𝑜c3o 7442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 df-tp 4130 df-suc 5646 df-1o 7447 df-2o 7448 df-3o 7449 |
This theorem is referenced by: (None) |
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