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Mirrors > Home > ILE Home > Th. List > elxpi | Unicode version |
Description: Membership in a cross product. Uses fewer axioms than elxp 4362. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
elxpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2046 | . . . . . 6 | |
2 | 1 | anbi1d 438 | . . . . 5 |
3 | 2 | 2exbidv 1748 | . . . 4 |
4 | 3 | elabg 2688 | . . 3 |
5 | 4 | ibi 165 | . 2 |
6 | df-xp 4351 | . . 3 | |
7 | df-opab 3819 | . . 3 | |
8 | 6, 7 | eqtri 2060 | . 2 |
9 | 5, 8 | eleq2s 2132 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cab 2026 cop 3378 copab 3817 cxp 4343 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-opab 3819 df-xp 4351 |
This theorem is referenced by: xpsspw 4450 dmaddpqlem 6475 nqpi 6476 enq0ref 6531 nqnq0 6539 nq0nn 6540 axaddcl 6940 axmulcl 6942 |
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