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Mirrors > Home > ILE Home > Th. List > difopab | Unicode version |
Description: The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Ref | Expression |
---|---|
difopab |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 4755 |
. . 3
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2 | reldif 4748 |
. . 3
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3 | 1, 2 | ax-mp 5 |
. 2
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4 | relopab 4755 |
. 2
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5 | sbcan 3007 |
. . . 4
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6 | sbcan 3007 |
. . . . 5
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7 | 6 | sbcbii 3024 |
. . . 4
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8 | opelopabsb 4262 |
. . . . 5
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9 | vex 2742 |
. . . . . . 7
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10 | sbcng 3005 |
. . . . . . 7
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11 | 9, 10 | ax-mp 5 |
. . . . . 6
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12 | vex 2742 |
. . . . . . . 8
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13 | sbcng 3005 |
. . . . . . . 8
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14 | 12, 13 | ax-mp 5 |
. . . . . . 7
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15 | 14 | sbcbii 3024 |
. . . . . 6
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16 | opelopabsb 4262 |
. . . . . . 7
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17 | 16 | notbii 668 |
. . . . . 6
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18 | 11, 15, 17 | 3bitr4ri 213 |
. . . . 5
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19 | 8, 18 | anbi12i 460 |
. . . 4
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20 | 5, 7, 19 | 3bitr4ri 213 |
. . 3
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21 | eldif 3140 |
. . 3
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22 | opelopabsb 4262 |
. . 3
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23 | 20, 21, 22 | 3bitr4i 212 |
. 2
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24 | 3, 4, 23 | eqrelriiv 4722 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-opab 4067 df-xp 4634 df-rel 4635 |
This theorem is referenced by: (None) |
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