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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 1925 | . . 3 | |
2 | vex 2733 | . . . . . 6 | |
3 | vex 2733 | . . . . . 6 | |
4 | 2, 3 | opex 4214 | . . . . 5 |
5 | eleq1 2233 | . . . . . 6 | |
6 | opelxp 4641 | . . . . . 6 | |
7 | 5, 6 | bitrdi 195 | . . . . 5 |
8 | 4, 7 | spcev 2825 | . . . 4 |
9 | 8 | exlimivv 1889 | . . 3 |
10 | 1, 9 | sylbir 134 | . 2 |
11 | elxp 4628 | . . . . 5 | |
12 | simpr 109 | . . . . . 6 | |
13 | 12 | 2eximi 1594 | . . . . 5 |
14 | 11, 13 | sylbi 120 | . . . 4 |
15 | 14 | exlimiv 1591 | . . 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 1348 wex 1485 wcel 2141 cop 3586 cxp 4609 |
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 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-opab 4051 df-xp 4617 |
This theorem is referenced by: xpm 5032 |
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