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Mirrors > Home > ILE Home > Th. List > 2dom | Unicode version |
Description: A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.) |
Ref | Expression |
---|---|
2dom |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df2o2 6431 |
. . . 4
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2 | 1 | breq1i 4010 |
. . 3
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3 | brdomi 6748 |
. . 3
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4 | 2, 3 | sylbi 121 |
. 2
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5 | f1f 5421 |
. . . . 5
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6 | 0ex 4130 |
. . . . . 6
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7 | 6 | prid1 3698 |
. . . . 5
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8 | ffvelcdm 5649 |
. . . . 5
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9 | 5, 7, 8 | sylancl 413 |
. . . 4
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10 | p0ex 4188 |
. . . . . 6
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11 | 10 | prid2 3699 |
. . . . 5
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12 | ffvelcdm 5649 |
. . . . 5
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13 | 5, 11, 12 | sylancl 413 |
. . . 4
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14 | 0nep0 4165 |
. . . . . 6
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15 | 14 | neii 2349 |
. . . . 5
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16 | f1fveq 5772 |
. . . . . 6
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17 | 7, 11, 16 | mpanr12 439 |
. . . . 5
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18 | 15, 17 | mtbiri 675 |
. . . 4
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19 | eqeq1 2184 |
. . . . . 6
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20 | 19 | notbid 667 |
. . . . 5
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21 | eqeq2 2187 |
. . . . . 6
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22 | 21 | notbid 667 |
. . . . 5
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23 | 20, 22 | rspc2ev 2856 |
. . . 4
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24 | 9, 13, 18, 23 | syl3anc 1238 |
. . 3
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25 | 24 | exlimiv 1598 |
. 2
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26 | 4, 25 | syl 14 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-suc 4371 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fv 5224 df-1o 6416 df-2o 6417 df-dom 6741 |
This theorem is referenced by: (None) |
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