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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1osng 5521 |
. . . . 5
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2 | 1 | ad2antrr 488 |
. . . 4
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3 | f1osng 5521 |
. . . . 5
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4 | 3 | ad2antlr 489 |
. . . 4
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5 | disjsn2 3670 |
. . . . 5
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6 | 5 | ad2antrl 490 |
. . . 4
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7 | disjsn2 3670 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 7 | ad2antll 491 |
. . . 4
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9 | f1oun 5500 |
. . . 4
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10 | 2, 4, 6, 8, 9 | syl22anc 1250 |
. . 3
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11 | df-pr 3614 |
. . . . . 6
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12 | 11 | eqcomi 2193 |
. . . . 5
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13 | 12 | a1i 9 |
. . . 4
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14 | df-pr 3614 |
. . . . . 6
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15 | 14 | eqcomi 2193 |
. . . . 5
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16 | 15 | a1i 9 |
. . . 4
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17 | df-pr 3614 |
. . . . . 6
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18 | 17 | eqcomi 2193 |
. . . . 5
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19 | 18 | a1i 9 |
. . . 4
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20 | 13, 16, 19 | f1oeq123d 5474 |
. . 3
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21 | 10, 20 | mpbid 147 |
. 2
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22 | 21 | ex 115 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 |
This theorem is referenced by: (None) |
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