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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 3884 | . . . 4 ⊢ ∅ ∈ V | |
3 | 1, 2 | xpsnen 6295 | . . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴 |
4 | endisj.2 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | 1on 6008 | . . . . 5 ⊢ 1𝑜 ∈ On | |
6 | 5 | elexi 2567 | . . . 4 ⊢ 1𝑜 ∈ V |
7 | 4, 6 | xpsnen 6295 | . . 3 ⊢ (𝐵 × {1𝑜}) ≈ 𝐵 |
8 | 3, 7 | pm3.2i 257 | . 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) |
9 | xp01disj 6017 | . 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅ | |
10 | p0ex 3939 | . . . 4 ⊢ {∅} ∈ V | |
11 | 1, 10 | xpex 4453 | . . 3 ⊢ (𝐴 × {∅}) ∈ V |
12 | 6 | snex 3937 | . . . 4 ⊢ {1𝑜} ∈ V |
13 | 4, 12 | xpex 4453 | . . 3 ⊢ (𝐵 × {1𝑜}) ∈ V |
14 | breq1 3767 | . . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴)) | |
15 | breq1 3767 | . . . . 5 ⊢ (𝑦 = (𝐵 × {1𝑜}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1𝑜}) ≈ 𝐵)) | |
16 | 14, 15 | bi2anan9 538 | . . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵))) |
17 | ineq12 3133 | . . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜}))) | |
18 | 17 | eqeq1d 2048 | . . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅)) |
19 | 16, 18 | anbi12d 442 | . . 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 2648 | . 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)) |
21 | 8, 9, 20 | mp2an 402 | 1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 ∩ cin 2916 ∅c0 3224 {csn 3375 class class class wbr 3764 Oncon0 4100 × cxp 4343 1𝑜c1o 5994 ≈ cen 6219 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-1o 6001 df-en 6222 |
This theorem is referenced by: (None) |
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