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Mirrors > Home > ILE Home > Th. List > elxpi | Unicode version |
Description: Membership in a cross product. Uses fewer axioms than elxp 4305. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
elxpi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2043 |
. . . . . 6
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2 | 1 | anbi1d 438 |
. . . . 5
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3 | 2 | 2exbidv 1745 |
. . . 4
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4 | 3 | elabg 2682 |
. . 3
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5 | 4 | ibi 165 |
. 2
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6 | df-xp 4294 |
. . 3
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7 | df-opab 3810 |
. . 3
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8 | 6, 7 | eqtri 2057 |
. 2
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9 | 5, 8 | eleq2s 2129 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-opab 3810 df-xp 4294 |
This theorem is referenced by: xpsspw 4393 dmaddpqlem 6361 nqpi 6362 enq0ref 6416 nqnq0 6424 nq0nn 6425 axaddcl 6750 axmulcl 6752 |
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