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Mirrors > Home > ILE Home > Th. List > endisj | GIF version |
Description: Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.) |
Ref | Expression |
---|---|
endisj.1 | ⊢ 𝐴 ∈ V |
endisj.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
endisj | ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endisj.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 0ex 4055 | . . . 4 ⊢ ∅ ∈ V | |
3 | 1, 2 | xpsnen 6715 | . . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴 |
4 | endisj.2 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | 1on 6320 | . . . . 5 ⊢ 1o ∈ On | |
6 | 5 | elexi 2698 | . . . 4 ⊢ 1o ∈ V |
7 | 4, 6 | xpsnen 6715 | . . 3 ⊢ (𝐵 × {1o}) ≈ 𝐵 |
8 | 3, 7 | pm3.2i 270 | . 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) |
9 | xp01disj 6330 | . 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅ | |
10 | p0ex 4112 | . . . 4 ⊢ {∅} ∈ V | |
11 | 1, 10 | xpex 4654 | . . 3 ⊢ (𝐴 × {∅}) ∈ V |
12 | 6 | snex 4109 | . . . 4 ⊢ {1o} ∈ V |
13 | 4, 12 | xpex 4654 | . . 3 ⊢ (𝐵 × {1o}) ∈ V |
14 | breq1 3932 | . . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴)) | |
15 | breq1 3932 | . . . . 5 ⊢ (𝑦 = (𝐵 × {1o}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1o}) ≈ 𝐵)) | |
16 | 14, 15 | bi2anan9 595 | . . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵))) |
17 | ineq12 3272 | . . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o}))) | |
18 | 17 | eqeq1d 2148 | . . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅)) |
19 | 16, 18 | anbi12d 464 | . . 3 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅))) |
20 | 11, 13, 19 | spc2ev 2781 | . 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)) |
21 | 8, 9, 20 | mp2an 422 | 1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 ∩ cin 3070 ∅c0 3363 {csn 3527 class class class wbr 3929 Oncon0 4285 × cxp 4537 1oc1o 6306 ≈ 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-in1 603 ax-in2 604 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-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 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-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 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-1o 6313 df-en 6635 |
This theorem is referenced by: (None) |
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