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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 5213 | . . . 4 ⊢ ∅ ∈ V | |
3 | 1, 2 | xpsnen 8603 | . . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴 |
4 | endisj.2 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | 1oex 8112 | . . . 4 ⊢ 1o ∈ V | |
6 | 4, 5 | xpsnen 8603 | . . 3 ⊢ (𝐵 × {1o}) ≈ 𝐵 |
7 | 3, 6 | pm3.2i 473 | . 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) |
8 | xp01disj 8122 | . 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅ | |
9 | p0ex 5287 | . . . 4 ⊢ {∅} ∈ V | |
10 | 1, 9 | xpex 7478 | . . 3 ⊢ (𝐴 × {∅}) ∈ V |
11 | snex 5334 | . . . 4 ⊢ {1o} ∈ V | |
12 | 4, 11 | xpex 7478 | . . 3 ⊢ (𝐵 × {1o}) ∈ V |
13 | breq1 5071 | . . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴)) | |
14 | breq1 5071 | . . . . 5 ⊢ (𝑦 = (𝐵 × {1o}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1o}) ≈ 𝐵)) | |
15 | 13, 14 | bi2anan9 637 | . . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵))) |
16 | ineq12 4186 | . . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o}))) | |
17 | 16 | eqeq1d 2825 | . . . 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 3610 | . 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)) |
20 | 7, 8, 19 | mp2an 690 | 1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3496 ∩ cin 3937 ∅c0 4293 {csn 4569 class class class wbr 5068 × cxp 5555 1oc1o 8097 ≈ cen 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-ord 6196 df-on 6197 df-suc 6199 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-1o 8104 df-en 8512 |
This theorem is referenced by: (None) |
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