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Mirrors > Home > ILE Home > Th. List > disjne | Unicode version |
Description: Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
disjne |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj 3381 | . . 3 | |
2 | eleq1 2180 | . . . . . 6 | |
3 | 2 | notbid 641 | . . . . 5 |
4 | 3 | rspccva 2762 | . . . 4 |
5 | eleq1a 2189 | . . . . 5 | |
6 | 5 | necon3bd 2328 | . . . 4 |
7 | 4, 6 | syl5com 29 | . . 3 |
8 | 1, 7 | sylanb 282 | . 2 |
9 | 8 | 3impia 1163 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 wne 2285 wral 2393 cin 3040 c0 3333 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-v 2662 df-dif 3043 df-in 3047 df-nul 3334 |
This theorem is referenced by: (None) |
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