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Mirrors > Home > MPE Home > Th. List > sdom1 | Structured version Visualization version Unicode version |
Description: A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
Ref | Expression |
---|---|
sdom1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnsym 7716 |
. . . . 5
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2 | 1 | con2i 124 |
. . . 4
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3 | 0sdom1dom 7788 |
. . . 4
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4 | 2, 3 | sylnibr 312 |
. . 3
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5 | relsdom 7594 |
. . . . 5
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6 | 5 | brrelexi 4880 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 0sdomg 7719 |
. . . . 5
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8 | 7 | necon2bbid 2686 |
. . . 4
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9 | 6, 8 | syl 17 |
. . 3
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10 | 4, 9 | mpbird 240 |
. 2
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11 | 1n0 7215 |
. . . 4
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12 | 1on 7207 |
. . . . . 6
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13 | 12 | elexi 3041 |
. . . . 5
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14 | 13 | 0sdom 7721 |
. . . 4
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15 | 11, 14 | mpbir 214 |
. . 3
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16 | breq1 4398 |
. . 3
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17 | 15, 16 | mpbiri 241 |
. 2
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18 | 10, 17 | impbii 192 |
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-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-om 6712 df-1o 7200 df-er 7381 df-en 7588 df-dom 7589 df-sdom 7590 |
This theorem is referenced by: modom 7791 frgpcyg 19221 |
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