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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elsngl | Structured version Unicode version |
Description: Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-elsngl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clel 2449 |
. 2
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2 | df-bj-sngl 32792 |
. . . . 5
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3 | 2 | abeq2i 2581 |
. . . 4
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4 | 3 | anbi2i 694 |
. . 3
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5 | 4 | exbii 1635 |
. 2
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6 | r19.42v 2981 |
. . . . 5
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7 | 6 | bicomi 202 |
. . . 4
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8 | 7 | exbii 1635 |
. . 3
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9 | rexcom4 3098 |
. . . 4
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10 | 9 | bicomi 202 |
. . 3
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11 | eqcom 2463 |
. . . . . 6
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12 | snex 4642 |
. . . . . . 7
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13 | 12 | eqvinc 3193 |
. . . . . 6
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14 | exancom 1639 |
. . . . . 6
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15 | 11, 13, 14 | 3bitri 271 |
. . . . 5
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16 | 15 | bicomi 202 |
. . . 4
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17 | 16 | rexbii 2862 |
. . 3
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18 | 8, 10, 17 | 3bitri 271 |
. 2
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19 | 1, 5, 18 | 3bitri 271 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4522 ax-nul 4530 ax-pr 4640 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-rex 2805 df-v 3080 df-dif 3440 df-un 3442 df-nul 3747 df-sn 3987 df-pr 3989 df-bj-sngl 32792 |
This theorem is referenced by: bj-snglc 32795 bj-snglss 32796 bj-0nelsngl 32797 bj-eltag 32803 |
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