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Mirrors > Home > ILE Home > Th. List > ssdomg | Unicode version |
Description: A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
ssdomg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 4157 |
. . 3
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2 | simpr 110 |
. . 3
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3 | f1oi 5518 |
. . . . . . . . . 10
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4 | dff1o3 5486 |
. . . . . . . . . 10
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5 | 3, 4 | mpbi 145 |
. . . . . . . . 9
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6 | 5 | simpli 111 |
. . . . . . . 8
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7 | fof 5457 |
. . . . . . . 8
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8 | 6, 7 | ax-mp 5 |
. . . . . . 7
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9 | fss 5396 |
. . . . . . 7
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10 | 8, 9 | mpan 424 |
. . . . . 6
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11 | funi 5267 |
. . . . . . . 8
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12 | cnvi 5051 |
. . . . . . . . 9
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13 | 12 | funeqi 5256 |
. . . . . . . 8
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14 | 11, 13 | mpbir 146 |
. . . . . . 7
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15 | funres11 5307 |
. . . . . . 7
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16 | 14, 15 | ax-mp 5 |
. . . . . 6
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17 | 10, 16 | jctir 313 |
. . . . 5
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18 | df-f1 5240 |
. . . . 5
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19 | 17, 18 | sylibr 134 |
. . . 4
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20 | 19 | adantr 276 |
. . 3
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21 | f1dom2g 6782 |
. . 3
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22 | 1, 2, 20, 21 | syl3anc 1249 |
. 2
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23 | 22 | expcom 116 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-dom 6768 |
This theorem is referenced by: cnvct 6835 ssct 6844 xpdom3m 6860 0domg 6865 mapdom1g 6875 phplem4dom 6890 nndomo 6892 phpm 6893 fict 6896 domfiexmid 6906 infnfi 6923 exmidfodomrlemr 7231 exmidfodomrlemrALT 7232 pw1dom2 7256 fihashss 10828 phicl2 12246 phibnd 12249 4sqlem11 12433 qnnen 12482 sbthom 15236 |
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