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| Mirrors > Home > MPE Home > Th. List > xp2dju | Structured version Visualization version GIF version | ||
| Description: Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| Ref | Expression |
|---|---|
| xp2dju | ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xpundir 5755 | . 2 ⊢ (({∅} ∪ {1o}) × 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
| 2 | df2o3 8514 | . . . 4 ⊢ 2o = {∅, 1o} | |
| 3 | df-pr 4629 | . . . 4 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
| 4 | 2, 3 | eqtri 2765 | . . 3 ⊢ 2o = ({∅} ∪ {1o}) |
| 5 | 4 | xpeq1i 5711 | . 2 ⊢ (2o × 𝐴) = (({∅} ∪ {1o}) × 𝐴) |
| 6 | df-dju 9941 | . 2 ⊢ (𝐴 ⊔ 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
| 7 | 1, 5, 6 | 3eqtr4i 2775 | 1 ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cun 3949 ∅c0 4333 {csn 4626 {cpr 4628 × cxp 5683 1oc1o 8499 2oc2o 8500 ⊔ cdju 9938 |
| 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-pr 4629 df-opab 5206 df-xp 5691 df-suc 6390 df-1o 8506 df-2o 8507 df-dju 9941 |
| This theorem is referenced by: pwdju1 10231 unctb 10244 infdjuabs 10245 ackbij1lem5 10263 fin56 10433 |
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