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Mirrors > Home > ILE Home > Th. List > opm | Unicode version |
Description: An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.) |
Ref | Expression |
---|---|
opm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-op 3541 |
. . . . 5
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2 | 1 | eleq2i 2207 |
. . . 4
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3 | 2 | exbii 1585 |
. . 3
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4 | abid 2128 |
. . . 4
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5 | 4 | exbii 1585 |
. . 3
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6 | 3, 5 | bitri 183 |
. 2
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7 | 19.42v 1879 |
. . 3
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8 | df-3an 965 |
. . . 4
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9 | 8 | exbii 1585 |
. . 3
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10 | df-3an 965 |
. . 3
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11 | 7, 9, 10 | 3bitr4ri 212 |
. 2
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12 | 3simpa 979 |
. . 3
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13 | id 19 |
. . . 4
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14 | snexg 4116 |
. . . . . 6
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15 | 14 | adantr 274 |
. . . . 5
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16 | prmg 3652 |
. . . . 5
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17 | 15, 16 | syl 14 |
. . . 4
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18 | 13, 17, 10 | sylanbrc 414 |
. . 3
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19 | 12, 18 | impbii 125 |
. 2
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20 | 6, 11, 19 | 3bitr2i 207 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 |
This theorem is referenced by: opnzi 4165 opeqex 4179 cnm 7664 setsfun0 12034 |
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