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Mirrors > Home > ILE Home > Th. List > exsnrex | 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2715 | . . . . . 6 | |
2 | 1 | snid 3591 | . . . . 5 |
3 | eleq2 2221 | . . . . 5 | |
4 | 2, 3 | mpbiri 167 | . . . 4 |
5 | 4 | pm4.71ri 390 | . . 3 |
6 | 5 | exbii 1585 | . 2 |
7 | df-rex 2441 | . 2 | |
8 | 6, 7 | bitr4i 186 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 wrex 2436 csn 3560 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 df-sn 3566 |
This theorem is referenced by: (None) |
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