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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 5743 | . 2 ⊢ (({∅} ∪ {1o}) × 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
2 | df2o3 8470 | . . . 4 ⊢ 2o = {∅, 1o} | |
3 | df-pr 4630 | . . . 4 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
4 | 2, 3 | eqtri 2760 | . . 3 ⊢ 2o = ({∅} ∪ {1o}) |
5 | 4 | xpeq1i 5701 | . 2 ⊢ (2o × 𝐴) = (({∅} ∪ {1o}) × 𝐴) |
6 | df-dju 9892 | . 2 ⊢ (𝐴 ⊔ 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
7 | 1, 5, 6 | 3eqtr4i 2770 | 1 ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∪ cun 3945 ∅c0 4321 {csn 4627 {cpr 4629 × cxp 5673 1oc1o 8455 2oc2o 8456 ⊔ cdju 9889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-dif 3950 df-un 3952 df-nul 4322 df-pr 4630 df-opab 5210 df-xp 5681 df-suc 6367 df-1o 8462 df-2o 8463 df-dju 9892 |
This theorem is referenced by: pwdju1 10181 unctb 10196 infdjuabs 10197 ackbij1lem5 10215 fin56 10384 |
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