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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 5030 | . . 3 | |
2 | anidm 394 | . . . . . 6 | |
3 | eleq2 2234 | . . . . . . . 8 | |
4 | 3 | exbidv 1818 | . . . . . . 7 |
5 | 4 | anbi2d 461 | . . . . . 6 |
6 | 2, 5 | bitr3id 193 | . . . . 5 |
7 | eqimss 3201 | . . . . . . . 8 | |
8 | ssxpbm 5044 | . . . . . . . 8 | |
9 | 7, 8 | syl5ibcom 154 | . . . . . . 7 |
10 | eqimss2 3202 | . . . . . . . 8 | |
11 | ssxpbm 5044 | . . . . . . . 8 | |
12 | 10, 11 | syl5ibcom 154 | . . . . . . 7 |
13 | 9, 12 | anim12d 333 | . . . . . 6 |
14 | an4 581 | . . . . . . 7 | |
15 | eqss 3162 | . . . . . . . 8 | |
16 | eqss 3162 | . . . . . . . 8 | |
17 | 15, 16 | anbi12i 457 | . . . . . . 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 4628 | . 2 | |
24 | 22, 23 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 wss 3121 cxp 4607 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-dm 4619 df-rn 4620 |
This theorem is referenced by: cc2lem 7217 |
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