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Mirrors > Home > ILE Home > Th. List > f1oprg | Unicode version |
Description: An unordered pair of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.) |
Ref | Expression |
---|---|
f1oprg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1osng 5401 | . . . . 5 | |
2 | 1 | ad2antrr 479 | . . . 4 |
3 | f1osng 5401 | . . . . 5 | |
4 | 3 | ad2antlr 480 | . . . 4 |
5 | disjsn2 3581 | . . . . 5 | |
6 | 5 | ad2antrl 481 | . . . 4 |
7 | disjsn2 3581 | . . . . 5 | |
8 | 7 | ad2antll 482 | . . . 4 |
9 | f1oun 5380 | . . . 4 | |
10 | 2, 4, 6, 8, 9 | syl22anc 1217 | . . 3 |
11 | df-pr 3529 | . . . . . 6 | |
12 | 11 | eqcomi 2141 | . . . . 5 |
13 | 12 | a1i 9 | . . . 4 |
14 | df-pr 3529 | . . . . . 6 | |
15 | 14 | eqcomi 2141 | . . . . 5 |
16 | 15 | a1i 9 | . . . 4 |
17 | df-pr 3529 | . . . . . 6 | |
18 | 17 | eqcomi 2141 | . . . . 5 |
19 | 18 | a1i 9 | . . . 4 |
20 | 13, 16, 19 | f1oeq123d 5357 | . . 3 |
21 | 10, 20 | mpbid 146 | . 2 |
22 | 21 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wne 2306 cun 3064 cin 3065 c0 3358 csn 3522 cpr 3523 cop 3525 wf1o 5117 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 |
This theorem is referenced by: (None) |
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