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Mirrors > Home > ILE Home > Th. List > en1 | Unicode version |
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) |
Ref | Expression |
---|---|
en1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 6013 | . . . . 5 | |
2 | 1 | breq2i 3772 | . . . 4 |
3 | bren 6228 | . . . 4 | |
4 | 2, 3 | bitri 173 | . . 3 |
5 | f1ocnv 5139 | . . . . 5 | |
6 | f1ofo 5133 | . . . . . . . 8 | |
7 | forn 5109 | . . . . . . . 8 | |
8 | 6, 7 | syl 14 | . . . . . . 7 |
9 | f1of 5126 | . . . . . . . . . 10 | |
10 | 0ex 3884 | . . . . . . . . . . . 12 | |
11 | 10 | fsn2 5337 | . . . . . . . . . . 11 |
12 | 11 | simprbi 260 | . . . . . . . . . 10 |
13 | 9, 12 | syl 14 | . . . . . . . . 9 |
14 | 13 | rneqd 4563 | . . . . . . . 8 |
15 | 10 | rnsnop 4801 | . . . . . . . 8 |
16 | 14, 15 | syl6eq 2088 | . . . . . . 7 |
17 | 8, 16 | eqtr3d 2074 | . . . . . 6 |
18 | 5, 17 | syl 14 | . . . . 5 |
19 | f1ofn 5127 | . . . . . . 7 | |
20 | 10 | snid 3402 | . . . . . . 7 |
21 | funfvex 5192 | . . . . . . . 8 | |
22 | 21 | funfni 4999 | . . . . . . 7 |
23 | 19, 20, 22 | sylancl 392 | . . . . . 6 |
24 | sneq 3386 | . . . . . . . 8 | |
25 | 24 | eqeq2d 2051 | . . . . . . 7 |
26 | 25 | spcegv 2641 | . . . . . 6 |
27 | 23, 26 | syl 14 | . . . . 5 |
28 | 5, 18, 27 | sylc 56 | . . . 4 |
29 | 28 | exlimiv 1489 | . . 3 |
30 | 4, 29 | sylbi 114 | . 2 |
31 | vex 2560 | . . . . 5 | |
32 | 31 | ensn1 6276 | . . . 4 |
33 | breq1 3767 | . . . 4 | |
34 | 32, 33 | mpbiri 157 | . . 3 |
35 | 34 | exlimiv 1489 | . 2 |
36 | 30, 35 | impbii 117 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 c0 3224 csn 3375 cop 3378 class class class wbr 3764 ccnv 4344 crn 4346 wfn 4897 wf 4898 wfo 4900 wf1o 4901 cfv 4902 c1o 5994 cen 6219 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-1o 6001 df-en 6222 |
This theorem is referenced by: en1bg 6280 reuen1 6281 |
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