Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4765 | . . 3 | |
2 | anidm 388 | . . . . . 6 | |
3 | eleq2 2142 | . . . . . . . 8 | |
4 | 3 | exbidv 1746 | . . . . . . 7 |
5 | 4 | anbi2d 451 | . . . . . 6 |
6 | 2, 5 | syl5bbr 192 | . . . . 5 |
7 | eqimss 3051 | . . . . . . . 8 | |
8 | ssxpbm 4776 | . . . . . . . 8 | |
9 | 7, 8 | syl5ibcom 153 | . . . . . . 7 |
10 | eqimss2 3052 | . . . . . . . 8 | |
11 | ssxpbm 4776 | . . . . . . . 8 | |
12 | 10, 11 | syl5ibcom 153 | . . . . . . 7 |
13 | 9, 12 | anim12d 328 | . . . . . 6 |
14 | an4 550 | . . . . . . 7 | |
15 | eqss 3014 | . . . . . . . 8 | |
16 | eqss 3014 | . . . . . . . 8 | |
17 | 15, 16 | anbi12i 447 | . . . . . . 7 |
18 | 14, 17 | bitr4i 185 | . . . . . 6 |
19 | 13, 18 | syl6ib 159 | . . . . 5 |
20 | 6, 19 | sylbid 148 | . . . 4 |
21 | 20 | com12 30 | . . 3 |
22 | 1, 21 | sylbi 119 | . 2 |
23 | xpeq12 4382 | . 2 | |
24 | 22, 23 | impbid1 140 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wss 2973 cxp 4361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |