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Mirrors > Home > ILE Home > Th. List > 0elsucexmid | Unicode version |
Description: If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
Ref | Expression |
---|---|
0elsucexmid.1 |
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Ref | Expression |
---|---|
0elsucexmid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtriexmidlem 4530 |
. . . 4
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2 | 0elsucexmid.1 |
. . . 4
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3 | suceq 4414 |
. . . . . 6
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4 | 3 | eleq2d 2257 |
. . . . 5
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5 | 4 | rspcv 2849 |
. . . 4
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6 | 1, 2, 5 | mp2 16 |
. . 3
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7 | 0ex 4142 |
. . . 4
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8 | 7 | elsuc 4418 |
. . 3
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9 | 6, 8 | mpbi 145 |
. 2
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10 | 7 | snid 3635 |
. . . . 5
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11 | biidd 172 |
. . . . . 6
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12 | 11 | elrab3 2906 |
. . . . 5
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13 | 10, 12 | ax-mp 5 |
. . . 4
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14 | 13 | biimpi 120 |
. . 3
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15 | ordtriexmidlem2 4531 |
. . . 4
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16 | 15 | eqcoms 2190 |
. . 3
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17 | 14, 16 | orim12i 760 |
. 2
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18 | 9, 17 | ax-mp 5 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-uni 3822 df-tr 4114 df-iord 4378 df-on 4380 df-suc 4383 |
This theorem is referenced by: (None) |
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