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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsneng 6709 | . . . . . . 7 | |
2 | 1 | 3adant2 1000 | . . . . . 6 |
3 | 2 | ensymd 6670 | . . . . 5 |
4 | xpexg 4648 | . . . . . . 7 | |
5 | 4 | 3adant3 1001 | . . . . . 6 |
6 | simp3 983 | . . . . . . . 8 | |
7 | 6 | snssd 3660 | . . . . . . 7 |
8 | xpss2 4645 | . . . . . . 7 | |
9 | 7, 8 | syl 14 | . . . . . 6 |
10 | ssdomg 6665 | . . . . . 6 | |
11 | 5, 9, 10 | sylc 62 | . . . . 5 |
12 | endomtr 6677 | . . . . 5 | |
13 | 3, 11, 12 | syl2anc 408 | . . . 4 |
14 | 13 | 3expia 1183 | . . 3 |
15 | 14 | exlimdv 1791 | . 2 |
16 | 15 | 3impia 1178 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wex 1468 wcel 1480 cvv 2681 wss 3066 csn 3522 class class class wbr 3924 cxp 4532 cen 6625 cdom 6626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-er 6422 df-en 6628 df-dom 6629 |
This theorem is referenced by: (None) |
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