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Mirrors > Home > ILE Home > Th. List > oprcl | Unicode version |
Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
oprcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex2 2753 |
. 2
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2 | df-op 3600 |
. . . . . . 7
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3 | 2 | eleq2i 2244 |
. . . . . 6
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4 | df-clab 2164 |
. . . . . 6
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5 | 3, 4 | bitri 184 |
. . . . 5
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6 | 3simpa 994 |
. . . . . 6
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7 | 6 | sbimi 1764 |
. . . . 5
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8 | 5, 7 | sylbi 121 |
. . . 4
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9 | nfv 1528 |
. . . . 5
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10 | 9 | sbf 1777 |
. . . 4
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11 | 8, 10 | sylib 122 |
. . 3
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12 | 11 | exlimiv 1598 |
. 2
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13 | 1, 12 | syl 14 |
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-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2739 df-op 3600 |
This theorem is referenced by: opth1 4232 opth 4233 0nelop 4244 |
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