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Mirrors > Home > ILE Home > Th. List > f1osng | Unicode version |
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.) |
Ref | Expression |
---|---|
f1osng |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3594 | . . . 4 | |
2 | f1oeq2 5432 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | opeq1 3765 | . . . . 5 | |
5 | 4 | sneqd 3596 | . . . 4 |
6 | f1oeq1 5431 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 3, 7 | bitrd 187 | . 2 |
9 | sneq 3594 | . . . 4 | |
10 | f1oeq3 5433 | . . . 4 | |
11 | 9, 10 | syl 14 | . . 3 |
12 | opeq2 3766 | . . . . 5 | |
13 | 12 | sneqd 3596 | . . . 4 |
14 | f1oeq1 5431 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | 11, 15 | bitrd 187 | . 2 |
17 | vex 2733 | . . 3 | |
18 | vex 2733 | . . 3 | |
19 | 17, 18 | f1osn 5482 | . 2 |
20 | 8, 16, 19 | vtocl2g 2794 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 csn 3583 cop 3586 wf1o 5197 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 |
This theorem is referenced by: f1sng 5484 f1oprg 5486 fsnunf 5696 dif1en 6857 1fv 10095 zfz1isolem1 10775 sumsnf 11372 prodsnf 11555 ennnfonelemhf1o 12368 |
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