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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 3695 | . . . . 5 | |
2 | orddi 809 | . . . . . 6 | |
3 | simpll 518 | . . . . . . 7 | |
4 | pm1.4 716 | . . . . . . . 8 | |
5 | 4 | ad2antll 482 | . . . . . . 7 |
6 | 3, 5 | jca 304 | . . . . . 6 |
7 | 2, 6 | sylbi 120 | . . . . 5 |
8 | 1, 7 | syl6bi 162 | . . . 4 |
9 | oranim 770 | . . . . . 6 | |
10 | df-ne 2307 | . . . . . . 7 | |
11 | df-ne 2307 | . . . . . . 7 | |
12 | 10, 11 | anbi12i 455 | . . . . . 6 |
13 | 9, 12 | sylnibr 666 | . . . . 5 |
14 | oranim 770 | . . . . . 6 | |
15 | df-ne 2307 | . . . . . . 7 | |
16 | df-ne 2307 | . . . . . . 7 | |
17 | 15, 16 | anbi12i 455 | . . . . . 6 |
18 | 14, 17 | sylnibr 666 | . . . . 5 |
19 | 13, 18 | anim12i 336 | . . . 4 |
20 | 8, 19 | syl6 33 | . . 3 |
21 | pm4.56 769 | . . 3 | |
22 | 20, 21 | syl6ib 160 | . 2 |
23 | 22 | necon2ad 2363 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 wceq 1331 wcel 1480 wne 2306 cpr 3523 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 |
This theorem is referenced by: (None) |
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