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Mirrors > Home > ILE Home > Th. List > xpmlem | Unicode version |
Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.) |
Ref | Expression |
---|---|
xpmlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eeanv 1902 | . . 3 | |
2 | vex 2684 | . . . . . 6 | |
3 | vex 2684 | . . . . . 6 | |
4 | 2, 3 | opex 4146 | . . . . 5 |
5 | eleq1 2200 | . . . . . 6 | |
6 | opelxp 4564 | . . . . . 6 | |
7 | 5, 6 | syl6bb 195 | . . . . 5 |
8 | 4, 7 | spcev 2775 | . . . 4 |
9 | 8 | exlimivv 1868 | . . 3 |
10 | 1, 9 | sylbir 134 | . 2 |
11 | elxp 4551 | . . . . 5 | |
12 | simpr 109 | . . . . . 6 | |
13 | 12 | 2eximi 1580 | . . . . 5 |
14 | 11, 13 | sylbi 120 | . . . 4 |
15 | 14 | exlimiv 1577 | . . 3 |
16 | 15, 1 | sylib 121 | . 2 |
17 | 10, 16 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cop 3525 cxp 4532 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 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-opab 3985 df-xp 4540 |
This theorem is referenced by: xpm 4955 |
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