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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 4645 | . 2 ⊢ (({∅} ∪ {1o}) × 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
2 | df2o3 6379 | . . . 4 ⊢ 2o = {∅, 1o} | |
3 | df-pr 3568 | . . . 4 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
4 | 2, 3 | eqtri 2178 | . . 3 ⊢ 2o = ({∅} ∪ {1o}) |
5 | 4 | xpeq1i 4608 | . 2 ⊢ (2o × 𝐴) = (({∅} ∪ {1o}) × 𝐴) |
6 | df-dju 6984 | . 2 ⊢ (𝐴 ⊔ 𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴)) | |
7 | 1, 5, 6 | 3eqtr4i 2188 | 1 ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∪ cun 3100 ∅c0 3395 {csn 3561 {cpr 3562 × cxp 4586 1oc1o 6358 2oc2o 6359 ⊔ cdju 6983 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-un 3106 df-nul 3396 df-pr 3568 df-opab 4028 df-suc 4333 df-xp 4594 df-1o 6365 df-2o 6366 df-dju 6984 |
This theorem is referenced by: (None) |
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