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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 6391 | . . . . 5 | |
2 | 1 | breq2i 3987 | . . . 4 |
3 | bren 6707 | . . . 4 | |
4 | 2, 3 | bitri 183 | . . 3 |
5 | f1ocnv 5442 | . . . . 5 | |
6 | f1ofo 5436 | . . . . . . . 8 | |
7 | forn 5410 | . . . . . . . 8 | |
8 | 6, 7 | syl 14 | . . . . . . 7 |
9 | f1of 5429 | . . . . . . . . . 10 | |
10 | 0ex 4106 | . . . . . . . . . . . 12 | |
11 | 10 | fsn2 5656 | . . . . . . . . . . 11 |
12 | 11 | simprbi 273 | . . . . . . . . . 10 |
13 | 9, 12 | syl 14 | . . . . . . . . 9 |
14 | 13 | rneqd 4830 | . . . . . . . 8 |
15 | 10 | rnsnop 5081 | . . . . . . . 8 |
16 | 14, 15 | eqtrdi 2213 | . . . . . . 7 |
17 | 8, 16 | eqtr3d 2199 | . . . . . 6 |
18 | 5, 17 | syl 14 | . . . . 5 |
19 | f1ofn 5430 | . . . . . . 7 | |
20 | 10 | snid 3604 | . . . . . . 7 |
21 | funfvex 5500 | . . . . . . . 8 | |
22 | 21 | funfni 5285 | . . . . . . 7 |
23 | 19, 20, 22 | sylancl 410 | . . . . . 6 |
24 | sneq 3584 | . . . . . . . 8 | |
25 | 24 | eqeq2d 2176 | . . . . . . 7 |
26 | 25 | spcegv 2812 | . . . . . 6 |
27 | 23, 26 | syl 14 | . . . . 5 |
28 | 5, 18, 27 | sylc 62 | . . . 4 |
29 | 28 | exlimiv 1585 | . . 3 |
30 | 4, 29 | sylbi 120 | . 2 |
31 | vex 2727 | . . . . 5 | |
32 | 31 | ensn1 6756 | . . . 4 |
33 | breq1 3982 | . . . 4 | |
34 | 32, 33 | mpbiri 167 | . . 3 |
35 | 34 | exlimiv 1585 | . 2 |
36 | 30, 35 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1342 wex 1479 wcel 2135 cvv 2724 c0 3407 csn 3573 cop 3576 class class class wbr 3979 ccnv 4600 crn 4602 wfn 5180 wf 5181 wfo 5183 wf1o 5184 cfv 5185 c1o 6371 cen 6698 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-id 4268 df-suc 4346 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-1o 6378 df-en 6701 |
This theorem is referenced by: en1bg 6760 reuen1 6761 pm54.43 7140 |
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