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Mirrors > Home > ILE Home > Th. List > snssb | Unicode version |
Description: Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.) |
Ref | Expression |
---|---|
snssb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3144 |
. 2
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2 | velsn 3609 |
. . . 4
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3 | 2 | imbi1i 238 |
. . 3
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4 | 3 | albii 1470 |
. 2
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5 | eleq1 2240 |
. . . . 5
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6 | 5 | pm5.74i 180 |
. . . 4
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7 | 6 | albii 1470 |
. . 3
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8 | 19.23v 1883 |
. . 3
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9 | isset 2743 |
. . . . 5
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10 | 9 | bicomi 132 |
. . . 4
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11 | 10 | imbi1i 238 |
. . 3
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12 | 7, 8, 11 | 3bitri 206 |
. 2
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13 | 1, 4, 12 | 3bitri 206 |
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-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-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-sn 3598 |
This theorem is referenced by: snssg 3726 |
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