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Mirrors > Home > MPE Home > Th. List > nnullss | Structured version Unicode version |
Description: A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.) |
Ref | Expression |
---|---|
nnullss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3746 |
. 2
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2 | vex 3073 |
. . . . 5
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3 | 2 | snss 4099 |
. . . 4
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4 | 2 | snnz 4093 |
. . . . 5
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5 | snex 4633 |
. . . . . 6
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6 | sseq1 3477 |
. . . . . . 7
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7 | neeq1 2729 |
. . . . . . 7
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8 | 6, 7 | anbi12d 710 |
. . . . . 6
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9 | 5, 8 | spcev 3162 |
. . . . 5
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10 | 4, 9 | mpan2 671 |
. . . 4
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11 | 3, 10 | sylbi 195 |
. . 3
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12 | 11 | exlimiv 1689 |
. 2
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13 | 1, 12 | sylbi 195 |
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 1952 ax-ext 2430 ax-sep 4513 ax-nul 4521 ax-pr 4631 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-v 3072 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-nul 3738 df-sn 3978 df-pr 3980 |
This theorem is referenced by: (None) |
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