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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 2626 |
. 2
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2 | df-op 3431 |
. . . . . . 7
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3 | 2 | eleq2i 2149 |
. . . . . 6
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4 | df-clab 2070 |
. . . . . 6
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5 | 3, 4 | bitri 182 |
. . . . 5
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6 | 3simpa 936 |
. . . . . 6
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7 | 6 | sbimi 1689 |
. . . . 5
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8 | 5, 7 | sylbi 119 |
. . . 4
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9 | nfv 1462 |
. . . . 5
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10 | 9 | sbf 1702 |
. . . 4
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11 | 8, 10 | sylib 120 |
. . 3
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12 | 11 | exlimiv 1530 |
. 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-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-v 2614 df-op 3431 |
This theorem is referenced by: opth1 4027 opth 4028 0nelop 4039 |
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