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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 5618 | . 2 ⊢ (({∅} ∪ {1o}) × 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
2 | df2o3 8217 | . . . 4 ⊢ 2o = {∅, 1o} | |
3 | df-pr 4544 | . . . 4 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
4 | 2, 3 | eqtri 2765 | . . 3 ⊢ 2o = ({∅} ∪ {1o}) |
5 | 4 | xpeq1i 5577 | . 2 ⊢ (2o × 𝐴) = (({∅} ∪ {1o}) × 𝐴) |
6 | df-dju 9517 | . 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 1543 ∪ cun 3864 ∅c0 4237 {csn 4541 {cpr 4543 × cxp 5549 1oc1o 8195 2oc2o 8196 ⊔ cdju 9514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-pr 4544 df-opab 5116 df-xp 5557 df-suc 6219 df-1o 8202 df-2o 8203 df-dju 9517 |
This theorem is referenced by: pwdju1 9804 unctb 9819 infdjuabs 9820 ackbij1lem5 9838 fin56 10007 |
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