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Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > sssneq | Unicode version |
Description: Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.) |
Ref | Expression |
---|---|
sssneq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 |
. . . . 5
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2 | simprl 521 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 1, 2 | sseldd 3125 |
. . . 4
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4 | elsni 3574 |
. . . 4
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5 | 3, 4 | syl 14 |
. . 3
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6 | simprr 522 |
. . . . 5
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7 | 1, 6 | sseldd 3125 |
. . . 4
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8 | elsni 3574 |
. . . 4
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9 | 7, 8 | syl 14 |
. . 3
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10 | 5, 9 | eqtr4d 2190 |
. 2
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11 | 10 | ralrimivva 2536 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1740 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ral 2437 df-v 2711 df-in 3104 df-ss 3111 df-sn 3562 |
This theorem is referenced by: (None) |
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