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Mirrors > Home > NFE Home > Th. List > endisj | GIF version |
Description: Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by set.mm contributors, 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 4110 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | complex 4104 | . . . 4 ⊢ ∼ ∅ ∈ V |
4 | 1, 3 | xpsnen 6049 | . . 3 ⊢ (A × { ∼ ∅}) ≈ A |
5 | endisj.2 | . . . 4 ⊢ B ∈ V | |
6 | 5, 2 | xpsnen 6049 | . . 3 ⊢ (B × {∅}) ≈ B |
7 | 4, 6 | pm3.2i 441 | . 2 ⊢ ((A × { ∼ ∅}) ≈ A ∧ (B × {∅}) ≈ B) |
8 | necompl 3544 | . . 3 ⊢ ∼ ∅ ≠ ∅ | |
9 | 3, 8 | xpnedisj 5513 | . 2 ⊢ ((A × { ∼ ∅}) ∩ (B × {∅})) = ∅ |
10 | snex 4111 | . . . 4 ⊢ { ∼ ∅} ∈ V | |
11 | 1, 10 | xpex 5115 | . . 3 ⊢ (A × { ∼ ∅}) ∈ V |
12 | snex 4111 | . . . 4 ⊢ {∅} ∈ V | |
13 | 5, 12 | xpex 5115 | . . 3 ⊢ (B × {∅}) ∈ V |
14 | breq1 4642 | . . . . 5 ⊢ (x = (A × { ∼ ∅}) → (x ≈ A ↔ (A × { ∼ ∅}) ≈ A)) | |
15 | breq1 4642 | . . . . 5 ⊢ (y = (B × {∅}) → (y ≈ B ↔ (B × {∅}) ≈ B)) | |
16 | 14, 15 | bi2anan9 843 | . . . 4 ⊢ ((x = (A × { ∼ ∅}) ∧ y = (B × {∅})) → ((x ≈ A ∧ y ≈ B) ↔ ((A × { ∼ ∅}) ≈ A ∧ (B × {∅}) ≈ B))) |
17 | ineq12 3452 | . . . . 5 ⊢ ((x = (A × { ∼ ∅}) ∧ y = (B × {∅})) → (x ∩ y) = ((A × { ∼ ∅}) ∩ (B × {∅}))) | |
18 | 17 | eqeq1d 2361 | . . . 4 ⊢ ((x = (A × { ∼ ∅}) ∧ y = (B × {∅})) → ((x ∩ y) = ∅ ↔ ((A × { ∼ ∅}) ∩ (B × {∅})) = ∅)) |
19 | 16, 18 | anbi12d 691 | . . 3 ⊢ ((x = (A × { ∼ ∅}) ∧ y = (B × {∅})) → (((x ≈ A ∧ y ≈ B) ∧ (x ∩ y) = ∅) ↔ (((A × { ∼ ∅}) ≈ A ∧ (B × {∅}) ≈ B) ∧ ((A × { ∼ ∅}) ∩ (B × {∅})) = ∅))) |
20 | 11, 13, 19 | spc2ev 2947 | . 2 ⊢ ((((A × { ∼ ∅}) ≈ A ∧ (B × {∅}) ≈ B) ∧ ((A × { ∼ ∅}) ∩ (B × {∅})) = ∅) → ∃x∃y((x ≈ A ∧ y ≈ B) ∧ (x ∩ y) = ∅)) |
21 | 7, 9, 20 | mp2an 653 | 1 ⊢ ∃x∃y((x ≈ A ∧ y ≈ B) ∧ (x ∩ y) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 ∩ cin 3208 ∅c0 3550 {csn 3737 class class class wbr 4639 × cxp 4770 ≈ cen 6028 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-2nd 4797 df-en 6029 |
This theorem is referenced by: (None) |
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