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Mirrors > Home > ILE Home > Th. List > xp2dju | 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 4668 | . 2 ⊢ (({∅} ∪ {1o}) × 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
2 | df2o3 6409 | . . . 4 ⊢ 2o = {∅, 1o} | |
3 | df-pr 3590 | . . . 4 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
4 | 2, 3 | eqtri 2191 | . . 3 ⊢ 2o = ({∅} ∪ {1o}) |
5 | 4 | xpeq1i 4631 | . 2 ⊢ (2o × 𝐴) = (({∅} ∪ {1o}) × 𝐴) |
6 | df-dju 7015 | . 2 ⊢ (𝐴 ⊔ 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
7 | 1, 5, 6 | 3eqtr4i 2201 | 1 ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∪ cun 3119 ∅c0 3414 {csn 3583 {cpr 3584 × cxp 4609 1oc1o 6388 2oc2o 6389 ⊔ cdju 7014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-nul 3415 df-pr 3590 df-opab 4051 df-suc 4356 df-xp 4617 df-1o 6395 df-2o 6396 df-dju 7015 |
This theorem is referenced by: (None) |
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