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| Mirrors > Home > MPE Home > Th. List > endisj | Structured version Visualization version GIF version | ||
| Description: Any two sets are equinumerous to two 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 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 3 | 1, 2 | xpsnen 9095 | . . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴 |
| 4 | endisj.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 5 | 1oex 8516 | . . . 4 ⊢ 1o ∈ V | |
| 6 | 4, 5 | xpsnen 9095 | . . 3 ⊢ (𝐵 × {1o}) ≈ 𝐵 |
| 7 | 3, 6 | pm3.2i 470 | . 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) |
| 8 | xp01disj 8529 | . 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅ | |
| 9 | p0ex 5384 | . . . 4 ⊢ {∅} ∈ V | |
| 10 | 1, 9 | xpex 7773 | . . 3 ⊢ (𝐴 × {∅}) ∈ V |
| 11 | snex 5436 | . . . 4 ⊢ {1o} ∈ V | |
| 12 | 4, 11 | xpex 7773 | . . 3 ⊢ (𝐵 × {1o}) ∈ V |
| 13 | breq1 5146 | . . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴)) | |
| 14 | breq1 5146 | . . . . 5 ⊢ (𝑦 = (𝐵 × {1o}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1o}) ≈ 𝐵)) | |
| 15 | 13, 14 | bi2anan9 638 | . . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵))) |
| 16 | ineq12 4215 | . . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o}))) | |
| 17 | 16 | eqeq1d 2739 | . . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅)) |
| 18 | 15, 17 | anbi12d 632 | . . 3 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅))) |
| 19 | 10, 12, 18 | spc2ev 3607 | . 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)) |
| 20 | 7, 8, 19 | mp2an 692 | 1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ∅c0 4333 {csn 4626 class class class wbr 5143 × cxp 5683 1oc1o 8499 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-1o 8506 df-en 8986 |
| This theorem is referenced by: (None) |
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