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Mirrors > Home > ILE Home > Th. List > prneimg | Unicode version |
Description: Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.) |
Ref | Expression |
---|---|
prneimg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq12bg 3544 | . . . . 5 | |
2 | orddi 733 | . . . . . 6 | |
3 | simpll 481 | . . . . . . 7 | |
4 | pm1.4 646 | . . . . . . . 8 | |
5 | 4 | ad2antll 460 | . . . . . . 7 |
6 | 3, 5 | jca 290 | . . . . . 6 |
7 | 2, 6 | sylbi 114 | . . . . 5 |
8 | 1, 7 | syl6bi 152 | . . . 4 |
9 | oranim 807 | . . . . . 6 | |
10 | df-ne 2206 | . . . . . . 7 | |
11 | df-ne 2206 | . . . . . . 7 | |
12 | 10, 11 | anbi12i 433 | . . . . . 6 |
13 | 9, 12 | sylnibr 602 | . . . . 5 |
14 | oranim 807 | . . . . . 6 | |
15 | df-ne 2206 | . . . . . . 7 | |
16 | df-ne 2206 | . . . . . . 7 | |
17 | 15, 16 | anbi12i 433 | . . . . . 6 |
18 | 14, 17 | sylnibr 602 | . . . . 5 |
19 | 13, 18 | anim12i 321 | . . . 4 |
20 | 8, 19 | syl6 29 | . . 3 |
21 | pm4.56 806 | . . 3 | |
22 | 20, 21 | syl6ib 150 | . 2 |
23 | 22 | necon2ad 2262 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 wceq 1243 wcel 1393 wne 2204 cpr 3376 |
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-in1 544 ax-in2 545 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-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 |
This theorem is referenced by: (None) |
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