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Mirrors > Home > ILE Home > Th. List > elqsn0m | Unicode version |
Description: An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.) |
Ref | Expression |
---|---|
elqsn0m |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2100 |
. 2
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2 | eleq2 2163 |
. . 3
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3 | 2 | exbidv 1764 |
. 2
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4 | eleq2 2163 |
. . . 4
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5 | 4 | biimpar 293 |
. . 3
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6 | ecdmn0m 6401 |
. . 3
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7 | 5, 6 | sylib 121 |
. 2
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8 | 1, 3, 7 | ectocld 6425 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 df-xp 4483 df-cnv 4485 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-ec 6361 df-qs 6365 |
This theorem is referenced by: elqsn0 6428 ecelqsdm 6429 |
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