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Mirrors > Home > ILE Home > Th. List > endisj | Unicode 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 | |
endisj.2 |
Ref | Expression |
---|---|
endisj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endisj.1 | . . . 4 | |
2 | 0ex 4050 | . . . 4 | |
3 | 1, 2 | xpsnen 6708 | . . 3 |
4 | endisj.2 | . . . 4 | |
5 | 1on 6313 | . . . . 5 | |
6 | 5 | elexi 2693 | . . . 4 |
7 | 4, 6 | xpsnen 6708 | . . 3 |
8 | 3, 7 | pm3.2i 270 | . 2 |
9 | xp01disj 6323 | . 2 | |
10 | p0ex 4107 | . . . 4 | |
11 | 1, 10 | xpex 4649 | . . 3 |
12 | 6 | snex 4104 | . . . 4 |
13 | 4, 12 | xpex 4649 | . . 3 |
14 | breq1 3927 | . . . . 5 | |
15 | breq1 3927 | . . . . 5 | |
16 | 14, 15 | bi2anan9 595 | . . . 4 |
17 | ineq12 3267 | . . . . 5 | |
18 | 17 | eqeq1d 2146 | . . . 4 |
19 | 16, 18 | anbi12d 464 | . . 3 |
20 | 11, 13, 19 | spc2ev 2776 | . 2 |
21 | 8, 9, 20 | mp2an 422 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wex 1468 wcel 1480 cvv 2681 cin 3065 c0 3358 csn 3522 class class class wbr 3924 con0 4280 cxp 4532 c1o 6299 cen 6625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-1o 6306 df-en 6628 |
This theorem is referenced by: (None) |
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