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Mirrors > Home > ILE Home > Th. List > xpdom3m | Unicode version |
Description: A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.) |
Ref | Expression |
---|---|
xpdom3m |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsneng 6667 |
. . . . . . 7
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2 | 1 | 3adant2 981 |
. . . . . 6
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3 | 2 | ensymd 6629 |
. . . . 5
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4 | xpexg 4611 |
. . . . . . 7
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5 | 4 | 3adant3 982 |
. . . . . 6
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6 | simp3 964 |
. . . . . . . 8
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7 | 6 | snssd 3629 |
. . . . . . 7
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8 | xpss2 4608 |
. . . . . . 7
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9 | 7, 8 | syl 14 |
. . . . . 6
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10 | ssdomg 6624 |
. . . . . 6
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11 | 5, 9, 10 | sylc 62 |
. . . . 5
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12 | endomtr 6636 |
. . . . 5
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13 | 3, 11, 12 | syl2anc 406 |
. . . 4
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14 | 13 | 3expia 1164 |
. . 3
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15 | 14 | exlimdv 1771 |
. 2
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16 | 15 | 3impia 1159 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-er 6381 df-en 6587 df-dom 6588 |
This theorem is referenced by: (None) |
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