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Mirrors > Home > MPE Home > Th. List > rabsnif | Structured version Visualization version Unicode version |
Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.) |
Ref | Expression |
---|---|
rabsnif.f |
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Ref | Expression |
---|---|
rabsnif |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabsnifsb 4031 |
. . 3
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2 | rabsnif.f |
. . . . 5
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3 | 2 | sbcieg 3288 |
. . . 4
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4 | 3 | ifbid 3894 |
. . 3
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5 | 1, 4 | syl5eq 2517 |
. 2
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6 | rab0 3756 |
. . . 4
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7 | ifid 3909 |
. . . 4
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8 | 6, 7 | eqtr4i 2496 |
. . 3
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9 | snprc 4027 |
. . . . 5
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10 | 9 | biimpi 199 |
. . . 4
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11 | rabeq 3024 |
. . . 4
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12 | 10, 11 | syl 17 |
. . 3
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13 | 10 | ifeq1d 3890 |
. . 3
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14 | 8, 12, 13 | 3eqtr4a 2531 |
. 2
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15 | 5, 14 | pm2.61i 169 |
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-or 377 df-an 378 df-3an 1009 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-ne 2643 df-rab 2765 df-v 3033 df-sbc 3256 df-dif 3393 df-un 3395 df-nul 3723 df-if 3873 df-sn 3960 |
This theorem is referenced by: suppsnop 6947 m1detdiag 19699 1loopgrvd2 39725 1hevtxdg1 39728 1egrvtxdg1 39732 |
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