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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 6326 | . . . . 5 | |
2 | 1 | breq2i 3937 | . . . 4 |
3 | bren 6641 | . . . 4 | |
4 | 2, 3 | bitri 183 | . . 3 |
5 | f1ocnv 5380 | . . . . 5 | |
6 | f1ofo 5374 | . . . . . . . 8 | |
7 | forn 5348 | . . . . . . . 8 | |
8 | 6, 7 | syl 14 | . . . . . . 7 |
9 | f1of 5367 | . . . . . . . . . 10 | |
10 | 0ex 4055 | . . . . . . . . . . . 12 | |
11 | 10 | fsn2 5594 | . . . . . . . . . . 11 |
12 | 11 | simprbi 273 | . . . . . . . . . 10 |
13 | 9, 12 | syl 14 | . . . . . . . . 9 |
14 | 13 | rneqd 4768 | . . . . . . . 8 |
15 | 10 | rnsnop 5019 | . . . . . . . 8 |
16 | 14, 15 | syl6eq 2188 | . . . . . . 7 |
17 | 8, 16 | eqtr3d 2174 | . . . . . 6 |
18 | 5, 17 | syl 14 | . . . . 5 |
19 | f1ofn 5368 | . . . . . . 7 | |
20 | 10 | snid 3556 | . . . . . . 7 |
21 | funfvex 5438 | . . . . . . . 8 | |
22 | 21 | funfni 5223 | . . . . . . 7 |
23 | 19, 20, 22 | sylancl 409 | . . . . . 6 |
24 | sneq 3538 | . . . . . . . 8 | |
25 | 24 | eqeq2d 2151 | . . . . . . 7 |
26 | 25 | spcegv 2774 | . . . . . 6 |
27 | 23, 26 | syl 14 | . . . . 5 |
28 | 5, 18, 27 | sylc 62 | . . . 4 |
29 | 28 | exlimiv 1577 | . . 3 |
30 | 4, 29 | sylbi 120 | . 2 |
31 | vex 2689 | . . . . 5 | |
32 | 31 | ensn1 6690 | . . . 4 |
33 | breq1 3932 | . . . 4 | |
34 | 32, 33 | mpbiri 167 | . . 3 |
35 | 34 | exlimiv 1577 | . 2 |
36 | 30, 35 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wex 1468 wcel 1480 cvv 2686 c0 3363 csn 3527 cop 3530 class class class wbr 3929 ccnv 4538 crn 4540 wfn 5118 wf 5119 wfo 5121 wf1o 5122 cfv 5123 c1o 6306 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-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-reu 2423 df-v 2688 df-sbc 2910 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-suc 4293 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-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1o 6313 df-en 6635 |
This theorem is referenced by: en1bg 6694 reuen1 6695 pm54.43 7046 |
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