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Mirrors > Home > ILE Home > Th. List > xpsneng | GIF version |
Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.) |
Ref | Expression |
---|---|
xpsneng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1 4553 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 × {𝑦}) = (𝐴 × {𝑦})) | |
2 | id 19 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | 1, 2 | breq12d 3942 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 × {𝑦}) ≈ 𝑥 ↔ (𝐴 × {𝑦}) ≈ 𝐴)) |
4 | sneq 3538 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵}) | |
5 | 4 | xpeq2d 4563 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 × {𝑦}) = (𝐴 × {𝐵})) |
6 | 5 | breq1d 3939 | . 2 ⊢ (𝑦 = 𝐵 → ((𝐴 × {𝑦}) ≈ 𝐴 ↔ (𝐴 × {𝐵}) ≈ 𝐴)) |
7 | vex 2689 | . . 3 ⊢ 𝑥 ∈ V | |
8 | vex 2689 | . . 3 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | xpsnen 6715 | . 2 ⊢ (𝑥 × {𝑦}) ≈ 𝑥 |
10 | 3, 6, 9 | vtocl2g 2750 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {csn 3527 class class class wbr 3929 × cxp 4537 ≈ cen 6632 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-en 6635 |
This theorem is referenced by: xp1en 6717 xpsnen2g 6723 xpdom3m 6728 hashxp 10572 pwf1oexmid 13194 |
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