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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 5705 | . 2 ⊢ (({∅} ∪ {1o}) × 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
2 | df2o3 8424 | . . . 4 ⊢ 2o = {∅, 1o} | |
3 | df-pr 4593 | . . . 4 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
4 | 2, 3 | eqtri 2761 | . . 3 ⊢ 2o = ({∅} ∪ {1o}) |
5 | 4 | xpeq1i 5663 | . 2 ⊢ (2o × 𝐴) = (({∅} ∪ {1o}) × 𝐴) |
6 | df-dju 9845 | . 2 ⊢ (𝐴 ⊔ 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
7 | 1, 5, 6 | 3eqtr4i 2771 | 1 ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cun 3912 ∅c0 4286 {csn 4590 {cpr 4592 × cxp 5635 1oc1o 8409 2oc2o 8410 ⊔ cdju 9842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-dif 3917 df-un 3919 df-nul 4287 df-pr 4593 df-opab 5172 df-xp 5643 df-suc 6327 df-1o 8416 df-2o 8417 df-dju 9845 |
This theorem is referenced by: pwdju1 10134 unctb 10149 infdjuabs 10150 ackbij1lem5 10168 fin56 10337 |
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