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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 5300 | . . . 4 ⊢ ∅ ∈ V | |
3 | 1, 2 | xpsnen 9054 | . . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴 |
4 | endisj.2 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | 1oex 8474 | . . . 4 ⊢ 1o ∈ V | |
6 | 4, 5 | xpsnen 9054 | . . 3 ⊢ (𝐵 × {1o}) ≈ 𝐵 |
7 | 3, 6 | pm3.2i 470 | . 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) |
8 | xp01disj 8489 | . 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅ | |
9 | p0ex 5375 | . . . 4 ⊢ {∅} ∈ V | |
10 | 1, 9 | xpex 7736 | . . 3 ⊢ (𝐴 × {∅}) ∈ V |
11 | snex 5424 | . . . 4 ⊢ {1o} ∈ V | |
12 | 4, 11 | xpex 7736 | . . 3 ⊢ (𝐵 × {1o}) ∈ V |
13 | breq1 5144 | . . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴)) | |
14 | breq1 5144 | . . . . 5 ⊢ (𝑦 = (𝐵 × {1o}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1o}) ≈ 𝐵)) | |
15 | 13, 14 | bi2anan9 636 | . . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵))) |
16 | ineq12 4202 | . . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o}))) | |
17 | 16 | eqeq1d 2728 | . . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅)) |
18 | 15, 17 | anbi12d 630 | . . 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 3591 | . 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)) |
20 | 7, 8, 19 | mp2an 689 | 1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3468 ∩ cin 3942 ∅c0 4317 {csn 4623 class class class wbr 5141 × cxp 5667 1oc1o 8457 ≈ cen 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-suc 6363 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-1o 8464 df-en 8939 |
This theorem is referenced by: (None) |
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