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Mirrors > Home > ILE Home > Th. List > xp11m | Unicode version |
Description: The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.) |
Ref | Expression |
---|---|
xp11m |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpm 4960 | . . 3 | |
2 | anidm 393 | . . . . . 6 | |
3 | eleq2 2203 | . . . . . . . 8 | |
4 | 3 | exbidv 1797 | . . . . . . 7 |
5 | 4 | anbi2d 459 | . . . . . 6 |
6 | 2, 5 | syl5bbr 193 | . . . . 5 |
7 | eqimss 3151 | . . . . . . . 8 | |
8 | ssxpbm 4974 | . . . . . . . 8 | |
9 | 7, 8 | syl5ibcom 154 | . . . . . . 7 |
10 | eqimss2 3152 | . . . . . . . 8 | |
11 | ssxpbm 4974 | . . . . . . . 8 | |
12 | 10, 11 | syl5ibcom 154 | . . . . . . 7 |
13 | 9, 12 | anim12d 333 | . . . . . 6 |
14 | an4 575 | . . . . . . 7 | |
15 | eqss 3112 | . . . . . . . 8 | |
16 | eqss 3112 | . . . . . . . 8 | |
17 | 15, 16 | anbi12i 455 | . . . . . . 7 |
18 | 14, 17 | bitr4i 186 | . . . . . 6 |
19 | 13, 18 | syl6ib 160 | . . . . 5 |
20 | 6, 19 | sylbid 149 | . . . 4 |
21 | 20 | com12 30 | . . 3 |
22 | 1, 21 | sylbi 120 | . 2 |
23 | xpeq12 4558 | . 2 | |
24 | 22, 23 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wss 3071 cxp 4537 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: (None) |
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