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Mirrors > Home > MPE Home > Th. List > exsnrex | Structured version Visualization version Unicode version |
Description: There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.) |
Ref | Expression |
---|---|
exsnrex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssnid 3989 |
. . . . 5
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2 | eleq2 2538 |
. . . . 5
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3 | 1, 2 | mpbiri 241 |
. . . 4
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4 | 3 | pm4.71ri 645 |
. . 3
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5 | 4 | exbii 1726 |
. 2
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6 | df-rex 2762 |
. 2
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7 | 5, 6 | bitr4i 260 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 |
This theorem depends on definitions: df-bi 190 df-an 378 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-rex 2762 df-v 3033 df-sn 3960 |
This theorem is referenced by: frgrawopreg1 25857 frgrawopreg2 25858 |
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