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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 7698 |
. . . . 5
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2 | 1 | con2i 124 |
. . . 4
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3 | 0sdom1dom 7770 |
. . . 4
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4 | 2, 3 | sylnibr 307 |
. . 3
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5 | relsdom 7576 |
. . . . 5
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6 | 5 | brrelexi 4875 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 0sdomg 7701 |
. . . . 5
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8 | 7 | necon2bbid 2667 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 6, 8 | syl 17 |
. . 3
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10 | 4, 9 | mpbird 236 |
. 2
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11 | 1n0 7197 |
. . . 4
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12 | 1on 7189 |
. . . . . 6
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13 | 12 | elexi 3055 |
. . . . 5
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14 | 13 | 0sdom 7703 |
. . . 4
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15 | 11, 14 | mpbir 213 |
. . 3
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16 | breq1 4405 |
. . 3
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17 | 15, 16 | mpbiri 237 |
. 2
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18 | 10, 17 | impbii 191 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-om 6693 df-1o 7182 df-er 7363 df-en 7570 df-dom 7571 df-sdom 7572 |
This theorem is referenced by: modom 7773 frgpcyg 19144 |
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