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Mirrors > Home > ILE Home > Th. List > en0 | Unicode version |
Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) |
Ref | Expression |
---|---|
en0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 6641 | . . 3 | |
2 | f1ocnv 5380 | . . . . 5 | |
3 | f1o00 5402 | . . . . . 6 | |
4 | 3 | simprbi 273 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | 5 | exlimiv 1577 | . . 3 |
7 | 1, 6 | sylbi 120 | . 2 |
8 | 0ex 4055 | . . . 4 | |
9 | 8 | enref 6659 | . . 3 |
10 | breq1 3932 | . . 3 | |
11 | 9, 10 | mpbiri 167 | . 2 |
12 | 7, 11 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 wceq 1331 wex 1468 c0 3363 class class class wbr 3929 ccnv 4538 wf1o 5122 cen 6632 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-en 6635 |
This theorem is referenced by: nneneq 6751 php5 6752 snnen2oprc 6754 php5dom 6757 ssfilem 6769 dif1enen 6774 fin0 6779 fin0or 6780 diffitest 6781 findcard 6782 findcard2 6783 findcard2s 6784 diffisn 6787 fiintim 6817 fisseneq 6820 fihasheq0 10540 zfz1iso 10584 |
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