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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 3572 | . . . 4 | |
2 | f1oeq2 5406 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | opeq1 3743 | . . . . 5 | |
5 | 4 | sneqd 3574 | . . . 4 |
6 | f1oeq1 5405 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 3, 7 | bitrd 187 | . 2 |
9 | sneq 3572 | . . . 4 | |
10 | f1oeq3 5407 | . . . 4 | |
11 | 9, 10 | syl 14 | . . 3 |
12 | opeq2 3744 | . . . . 5 | |
13 | 12 | sneqd 3574 | . . . 4 |
14 | f1oeq1 5405 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | 11, 15 | bitrd 187 | . 2 |
17 | vex 2715 | . . 3 | |
18 | vex 2715 | . . 3 | |
19 | 17, 18 | f1osn 5456 | . 2 |
20 | 8, 16, 19 | vtocl2g 2776 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 csn 3561 cop 3564 wf1o 5171 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 |
This theorem is referenced by: f1sng 5458 f1oprg 5460 fsnunf 5669 dif1en 6826 1fv 10047 zfz1isolem1 10722 sumsnf 11317 prodsnf 11500 ennnfonelemhf1o 12212 |
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