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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eu2ndop1stv | Structured version Visualization version Unicode version |
Description: If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
Ref | Expression |
---|---|
eu2ndop1stv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euex 2323 |
. 2
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2 | nfeu1 2309 |
. . . 4
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3 | nfcv 2592 |
. . . . 5
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4 | 3 | nfel1 2606 |
. . . 4
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5 | 2, 4 | nfim 2003 |
. . 3
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6 | opprc1 4189 |
. . . . . . . . 9
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7 | 6 | eleq1d 2513 |
. . . . . . . 8
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8 | ax-5 1758 |
. . . . . . . . 9
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9 | alneu 38622 |
. . . . . . . . 9
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10 | 8, 9 | syl 17 |
. . . . . . . 8
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11 | 7, 10 | syl6bi 232 |
. . . . . . 7
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12 | 11 | impcom 432 |
. . . . . 6
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13 | 7 | eubidv 2319 |
. . . . . . . 8
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14 | 13 | notbid 296 |
. . . . . . 7
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15 | 14 | adantl 468 |
. . . . . 6
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16 | 12, 15 | mpbird 236 |
. . . . 5
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17 | 16 | ex 436 |
. . . 4
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18 | 17 | con4d 109 |
. . 3
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19 | 5, 18 | exlimi 1995 |
. 2
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20 | 1, 19 | mpcom 37 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-nul 4534 ax-pow 4581 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-v 3047 df-dif 3407 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-op 3975 |
This theorem is referenced by: afveu 38655 tz6.12-afv 38675 |
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